Vectors - Skylight Publishing · to vectors. In physics and other applications, it is often...

1

4 Vectors

4.1 Prologue A vector is an entity that is characterized both by its magnitude and its direction. Vectors are widely used in physics. For example, if you want to describe the motion of an object at any given point in time, you might be interested in its speed (how fast it is going) and the direction of the motion. The velocity vector is a convenient way to combine these two characteristics (Figure 4-1-a). Similarly, when a force is applied to an object, it is convenient to represent it as a vector, because a force is characterized both by its magnitude and its direction. Figure 4-1-b shows the three forces that act on a block resting on a wedge: the gravitational force, the normal force, and the friction force. A vector can also represent the displacement of an object: if an object has moved from point A to point B, the arrow that connects A to B is called the displacement vector (Figure 4-1-c).

v

F

G

N

(a) (b)

A

B

d

(c)

Figure 4-1. (a) velocity of an object is a vector

(b) force is a vector (c) displacement is a vector

Some quantities are characterized only by their magnitude and have no direction. Such directionless quantities are called scalars, as opposed to vectors. The mass, temperature, or energy of an object are examples of scalars. In diagrams that involve vectors, like Figure 4-1, a vector is represented as an arrow; its magnitude is represented by the length of the arrow, and its direction is represented by the direction of the arrow. In these pictures, the vectors are two-

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2 CHAPTER 4 ~ VECTORS

dimensional. We can also easily envision vectors in three dimensions, and we can talk about vectors in four-, five-, or n-dimensional space. In mathematics, the concept of a vector is more abstract. A two-dimensional vector is simply an ordered pair of real numbers, (x, y). A three-dimensional vector is an ordered triplet of numbers, (x, y, z). An n-dimensional vector is an n-tuple, that is, an ordered sequence of n numbers, (x1, x2, ..., xn). Representing vectors using numbers is akin to representing points by their coordinates. This is convenient in formal computations and mathematical models. Vectors are used in many areas: physics, statistics, and engineering, to name a few. The branch of mathematics that studies n-dimensional vectors and operations on them is called linear algebra. In this chapter we will discuss how the geometrical concept of a vector (“an arrow”) relates to the more abstract algebraic concept (an ordered sequence of n numbers), and we will see which arithmetic operations on vectors make sense and how they are defined.

4.2 Geometric Representation of Vectors To keep things simple for now, we will consider only two-dimensional vectors. Let us imagine one “universal” plane. A two-dimensional vector v can be represented geometrically as an arrow on that plane:

v

The length of the arrow represents the magnitude of the vector; the direction of the arrow represents the direction of the vector.

The length of the vector v is denoted by v .

There is no generally accepted notation for the direction of a vector.

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CHAPTER 4 ~ VECTORS 3

You might be wondering: And what does the position of the arrow represent? The answer is: nothing! The position is irrelevant.

Two arrows that have the same length and the same direction represent the same vector, regardless of where the arrows are drawn on the plane.

So if we shift an arrow parallel to itself, it still represents the same vector:

This may seem counterintuitive. In charts and diagrams, a vector is often attached to a particular location. For example, we might draw the velocity vector for a moving particle at a given point on its path, or we can see vectors representing the speed and direction of wind at different geographic locations on a weather chart. In these situations, however, the vector depends on location, but it does not mean that a vector itself has a location. After all, when we see numbers that represent temperatures at different geographic locations on a weather chart we don’t conclude that a number has a location. In mathematics, the decision whether to call certain objects equivalent depends on a particular theory or branch of mathematics. When you studied arithmetic, for example, you were not surprised that “5” written in different places on the blackboard represented the same number. When you studied geometry, you learned the concept of congruent figures, which have the same size and shape but can have different positions and orientations. For the purpose of geometry, congruent figures are considered equivalent. In topology, the branch of mathematics that studies continuous transformations of shapes, the exact shape of a figure is not important: two figures are considered equivalent if one can be transformed into the other by a continuous transformation without ruptures or gluing. (In real life, too, things that are considered the same in one situation may count as different in another. If you are buying bubble gum, $1.00 and four quarters represent the same amount, but if you are need coins for a parking meter, one dollar bill is not the same as four quarters.) For vectors, the magnitude (length) and orientation are important, but the position is not.

A relation between objects that defines which two objects are deemed “equal” or “equivalent” is called an equivalence relation.

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As we said, an equivalence relation can be introduced in a particular mathematical theory if it is meaningful and useful. A proper equivalence relation must always have the following three properties: Reflexivity: any object is equivalent to itself.

Symmetry: If a is equivalent to b then b is equivalent to a.

Transitivity: If a is equivalent to b, and b is equivalent to c, then a is equivalent to c. The reflexivity, symmetry, and transitivity properties of an equivalence relation defined for elements of a set ensure that all the elements in the set fall neatly into non-overlapping classes (subsets) of equivalent elements: any two elements in the same class are equivalent to each other, and two elements from different classes are not equivalent.

Example 1 Recall that two triangles on the plane are called similar if they have the same shape but not necessarily the same size, that is, T1 is similar to T2 if T1 is congruent to T2 stretched or shrunk by some positive factor. Show that similarity of triangles is an equivalence relation. Solution We need to show that similarity of triangles has the reflexivity, symmetry, and transitivity properties. Reflexivity: any triangle T is congruent to itself, “stretched” by a factor of 1.

Symmetry: if T1 is congruent to T2, stretched or shrunk by a factor of k, then T2 is

congruent to T1, stretched or shrunk by a factor of 1k

.

Transitivity: if T1 is congruent to T2, stretched by a factor of k, and T2 is congruent to T3, stretched by a factor of m, then T1 is congruent to T3, stretched by a factor of k m⋅ .

Questions <...> - <...> in the exercises provide other examples of equivalence relations. In particular, calling two arrows (on the plane or in three-dimensional space) equal if they have the same lengths and directions defines an equivalence

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relation on the set of all arrows. The equivalence classes in this relation correspond to vectors.

In physics and other applications, it is often necessary to know the angle between two vectors or to take a projection of a vector onto a given line. The projection is a new vector located on the line (or directed along the line). In Figure 4-1-b, for example, the block is at rest, so all the forces acting on it must balance each other. If we take the projection of the gravitational force G onto the line of F , that projection must balance out F , so it must be equal in length and point in the opposite direction. If we take the projection of G onto the line of N , that projection must balance out N :

G

N

F

To get the projection of a vector v onto a line l, draw the vector somewhere on the plane, from the beginning point B to the endpoint E. Draw lines perpendicular to l from B and E. The points where these lines intersect l are the beginning and the end, respectively, of the projection vector p :

p

v l

B

E

Note that if we shift v , its projection p might shift, too, but will remain the same vector. If we shift the line parallel to itself, the projection vector does not change. If we choose a point O on a line and draw v from that point, then we can consider the angle θ between v and the line:

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θ v

O

V

P p

l

If θ is an acute angle, OV, VP, and OP form a right triangle with the hypotenuse OV and the legs VP and OP. Recall that in a right triangle, the ratio of the length of the leg adjacent to the angle θ to the length of the hypotenuse is called cosine of θ :

cospOP

OV vθ= =

So cosp v θ= . There is a special case when v is parallel to the line. In this case, θ is equal to 0, and p is equal to v . Another special case is when v is perpendicular to the line. In

this case, θ is equal to 902π

° = , and p is equal to 0. In these two cases the right

triangle is “squashed” to one dimension, but the formula cosp v θ= still holds, if we define cosθ to be equal to 1 when 0θ = , and define cosθ to be equal to 0 when

2πθ = .

cos 0 1=

cos 02π=

If the line of projection has a direction defined, (for example, if it is itself a vector), then the projection vector p may point in the same “positive” direction as the line or in the opposite “negative” direction:

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θ p

v

O

V

P θ

p

v

O

V

P

In the latter case, the angle θ is obtuse. It makes sense to define its cosine as a negative number, because p points in the negative direction. When θ is obtuse, we

define cosθ as cospOP

OV vθ = − = − . In particular, when 180θ π= ° = , cos 1θ = − .

cos 1π = −

Figure 4-2 summarizes the values of cosθ for different angles.

θ cos0 1=

cos 02π=

0 cos 1θ< < 1 cos 0θ− < <

cos 1π = −

Figure 4-2. The values of cosθ for different angles

Example 2

40°

120 lbs

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A rope holds back a 120-pound sled from sliding down a frictionless 40-degree slope. The line of the rope is parallel to the slope. What is the pull force of the rope? Solution The pull force F must counterbalance the projection of the gravitational force G onto the line of the slope:

40°

G = 120 lbs

F

A

B

O

In triangle AOB, 40 , 90 40 50ABO AOB= ° = ° − ° = ° . cos50AO OB= ° . Setting our calculator to degrees mode, we determine cos50 0.643° ≈ (or, setting our

calculator to radians mode, we determine 50cos 0.643180π ⋅

≈ ). The magnitude of the

pull force is 120 0.643 77.135⋅ ≈ lbs.

Exercises 1. The figure below shows several vectors.

a b

c d

e f

Are any two of them equal? If so, which ones?

2. Which of the following quantities are scalars: speed, force, energy,

momentum, weight, altitude, atmospheric pressure, distance, compass direction? Hint: If you’re not sure how the quantity is defined, look it up on the Internet.

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3. A particle is moving counterclockwise along a circle at a constant speed. The arrow in the figure below shows its velocity vector at a point A. Draw the particle’s velocity vectors at points B, C, D, and E.

A

C

D

B

E 4. LP (“long-playing”) records were introduced in the late 1940s. They had a

diameter of 10 or 12 inches and spun at 33⅓ rpm (revolutions per minute). If the diameter of a record is 12 inches, what is the length of the velocity vector, measured in meters per second, of a point on the edge of the record when the record is playing? Hint: Look at the the distance that a point on the edge covers in one minute.

5. Let us declare two arrows on the plane equivalent if they have the same

length and direction. Is this a proper equivalence relation? Explain. 6. Let us declare two arrows on the plane equivalent if they have the same

direction. Is this a properly defined equivalence relation? Explain why or why not.

7. Let us call two arrows on the plane equivalent if they have the same direction

or the same length. Is this a properly defined equivalence relation? Explain why or why not.

8. Let us call two points on the coordinate plane equivalent if their distances

from the given point O are the same. Show that this is an equivalence relation and describe the equivalence classes for it.

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9. A weight is swinging on a string like a pendulum:

Draw the gravitational force and the string pull force vectors acting on the weight. Make sure the projection of the gravitational force onto the line of the string balances out the pull force.

10. For each of the vectors shown below, draw its projection onto a horizontal

line.

11. For each of the vectors in Question 10, state its length and the cosine of the

angle that the vector forms with the horizontal line, assuming that the line’s positive direction is to the right.

12. In Figure 4-1-b the projection of the gravitational force G onto the normal

line (the line perpendicular to the surface) counterbalances the normal force N . If the slope of the wedge is 24 degrees and the weight of the block is 5 kg, what is the magnitude, in kilograms, of the normal force? Hint: Convert degrees into radians or set your calculator to degrees mode.

13. Under the conditions described in Question 12, what is the magnitude, in

kilograms, of the friction force F ? 14. After a hit, a baseball is launched at 30 degrees to the ground with an initial

speed of 55 miles per hour. The ball lands 350 feet away from the batter. For how many seconds did the ball fly? Assume that the horizontal component of the ball’s velocity remains constant. Hint: 1 mile = 5280 feet.

15. Is it always possible to measure the angle between two three-dimensional

vectors of non-zero lengths?

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4.3 Addition, Subtraction, and Multiplication by a Number

Vectors would be not very useful if we couldn’t perform some arithmetic operations on them. The sum of two vectors is defined as follows. First we shift the vectors so that they are drawn from the same point O. Then we draw the parallelogram using the two vectors as its adjacent sides. The diagonal of the parallelogram from O to the opposite vertex is the sum of the vectors (Figure 4-3). This is called the parallelogram method (or parallelogram rule).

u

v u v+

O

Figure 4-3. The parallelogram method for adding two vectors

There is a special case when the vectors lie on the same line (that is when they have the same or opposite directions). In this case, the “parallelogram” is squashed into a one-dimensional shape, but we can still determine the location of the “vertex” opposite to O:

u

v O u

v u v+ O

u v+

Another way to add two vectors is called the triangle method. In this method we draw the first vector from O to P and the second vector from P. The vector that connects O with the end of the second vector is their sum (Figure 4-4).

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u

v u v+

O P

Figure 4-4. The triangle method for adding two vectors

Naturally, the parallelogram method and the triangle method give the same result (Figure 4-5).

u

v u v+

O

v

Figure 4-5. The parallelogram method and the triangle

method for adding two vectors give the same result

Note that in both addition methods, the sum is drawn from a particular point. This is useful for getting the sum, but, just like any vector, the sum could really be drawn from any point — the position doesn’t matter. The above definition of addition matches our knowledge about the physical world. For example, imagine a tug-of-war with two people pulling in different directions. If two forces are applied to an object, the total force, their sum, can be found by adding the corresponding vectors, using the parallelogram or the triangle method. Also, if we consider only vectors on the x-axis, then the addition on them corresponds to the addition of numbers.

Subtraction of vectors is inverse to addition: if u v d− = then v d u+ = . Thus the difference of two vectors is the other diagonal of the parallelogram (Figure 4-6).

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u

v

O

u v−

Figure 4-6. The difference of two vectors; ( )v u v u+ − =

You might ask: What is v v− ? “The other diagonal” does not exist — it is just a point. To preserve consistency, it makes sense to define a zero vector. The zero vector is a vector of zero length, with undefined direction. (This is similar to the origin in polar coordinates: its radius is zero, and its argument is undefined.) The zero vector is denoted simply by 0. This is not the same as the number 0: the former is a vector, the latter is a scalar (a number). It is usually clear from the context which one of them is meant. We might sometimes write the zero vector as 0 , when we need to disambiguate the two.

0v v+ = , for any vector v .

The addition of vectors obeys the same laws as the addition of numbers: Commutativity: for any vectors u and v , u v v u+ = + .

Associativity: for any vectors u , v , and w , ( ) ( )u v w u v w+ + = + + .

Existence of zero: There is one vector 0, such that 0v v+ = , for any vector v .

Additive inverse: for any vector v there is a unique vector v− such that ( ) 0v v+ − = . ( v− has the same length as v and the opposite

direction.)

What about multiplication? Here things get a little more complicated. Let us first discuss what it means to multiply a vector by a number. We will return to the question of multiplying two vectors in the next section.

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If v is a vector and k is a positive real number, their product kv is the vector whose direction is the same as the direction of v and whose length is k times the length of v . We basically just stretch (or shrink) v by a factor of k (Figure 4-7-a). This makes multiplication consistent with addition. For example, we need to have 3v v v v= + + . When k is negative, we stretch (or shrink) the vector k times and change its direction to the opposite one (Figure 4-7-b). This makes sense, because we must have ( )1 v v− ⋅ = − . Finally, 0 0v = and 0 0k = .

(a) (b)

v

3v

1.5v v

3v−

1.5v−

0.5v 0.5v−

Figure 4-7. Multiplication of a vector by a number

Multiplication of a vector by a number also satisfies the distributive law: Distributive law: ( )k u v ku kv+ = +

If two operations — addition and multiplication by a number — are defined for the elements of a set, and if these operations have all the “nice” properties described above (commutativity and associativity of addition, existence of 0 and additive inverse, and distributive law for multiplication by a number), such a set is called a vector space over real numbers.

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Exercises 1. For a given pair of vectors u and v draw u v+ :

(c)

v

u

(b)

v

u

(a)

v

u

(f)

v u

(e)

v u

(d)

v

u

2. Draw u v− for each pair of vectors u and v in Question 1. 3. Draw 1.5 0.5u v− for each pair of vectors u and v in Question 1. 4. Under what conditions is it true that u v u v= ? 5. Show that for any vectors u and v , the projection of their sum u v+ onto a

line is equal to the sum of the projections of u and v onto that line. 6. Is it true that for any vectors u and v , u v u v+ ≥ + ? Explain. 7. Let us define the distance between vectors u and v as ( , )d u v v u= − .

Show that ( , )d u v has the three required properties of distance: ( , ) 0d u u = , ( , ) ( , )d u v d v u= , and the triangle rule (see Section <...>).

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8.

A

B C

The vectors AB , BC , and CA form a triangle, “traced” counterclockwise. Is it true that the sum of three such vectors is always 0?

9. In Question 8, the three vectors form a closed circuit of three vectors: the end

of a vector is the beginning point of the next one. Suppose we now make a closed circuit of n vectors. For example:

Is it true that the sum of all the vectors in the circuit is always 0?

10. Answer Question 9 for vectors in the three-dimensional space. 11. Show that the sum of the five vectors that connect the center of a regular

pentagon to its vertices is equal to 0. Is it true for any regular polygon? 12. Consider all quadratic functions 2( ) , ( 0)f x ax bx c a= + + ≠ . Do they form

a vector space (with addition and multiplication by a number defined in the usual way)? If so, guess how many dimensions this space has.

13. Consider together all constants, linear, and quadratic functions (that is, all the

functions of the form 2( )f x ax bx c= + + , where a, b, and c can be zeroes). Do they form a vector space (with addition and multiplication by a number defined in the usual way)? If so, guess how many dimensions this space has.

14. Consider all arithmetic sequences of real numbers. Let us define

1 2 1 2 1 1 2 2( , ,..., ,...) ( , ,..., ,...) ( , ,..., ,...)n n n na a a b b b a b a b a b+ = + + + and

1 2 1 2( , ,..., ,...) ( , ,..., ,...)n nk a a a ka ka ka= . Is this set of sequences with these operations a vector space over real numbers? If so, guess how many dimensions this space has.

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4.4 Dot Product There are several ways to define the product of two vectors. Each has its place and its limitations. Here we are concerned only with one operation, called the dot product.

The dot product of u and v is denoted as u v⋅ . Hence the name.

The dot-product of two vectors is not a vector: it is a number (a scalar). That is why the dot product is sometimes called the scalar product.

u v⋅ is equal to cosu v θ⋅ ⋅ , where θ is the angle between the vectors u and v .

Example 1

u

v

u is drawn on the side of a square with side 1, and v is drawn on the diagonal of the square. Calculate u v⋅ . Solution

1cos 45 1 2 12

u v u v⋅ = ° = ⋅ ⋅ = .

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Example 2

u

v

u and v are drawn on the sides of an equilateral triangle with side 3. Calculate u v⋅ . Solution

1cos60 3 3 4.52

u v u v⋅ = ° = ⋅ ⋅ = .

If u and v have the same direction, then u v⋅ is equal to u v⋅ . In particular,

2v v v⋅ = ; v v v= ⋅

If u and v have opposite directions, then u v u v⋅ = − ⋅ . For any vector v , 0 0 0v v⋅ = ⋅ = .

0u v⋅ > when u and v form an acute angle.

0u v⋅ = if and only if u and v are perpendicular to each other (or 0u = or 0v = ).

0u v⋅ < when u and v form an obtuse angle. If we know the lengths of two vectors and their dot product, we can find the angle between them. In Section <...> we will see that there is a simple formula for the dot product of vectors with known coordinates, so we can easily calculate cosθ and then use a calculator to find θ .

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The dot product has some of the “nice” properties of multiplication: Commutativity: for any vectors u and v , u v v u⋅ = ⋅ .

Associativity with multiplication by a number: for any number k and vectors u , v , ( ) ( )ku v k u v⋅ = ⋅ .

Distributive law: ( )u v w u w v w+ ⋅ = ⋅ + ⋅ . The first two properties are pretty obvious (see Questions <...> and <...> in the exercises). The proof of the distributive law requires a couple of steps. It is not hard to prove the distributive law for three parallel vectors (see Question <...> in the exercises). To prove the general distributive law, note that

wu w u w⋅ = ⋅ , where wu is the projection of u onto w (see Question <...> in the exercises). Likewise, wv w v w⋅ = ⋅ . Also, if ( )wu v+ is the projection of u v+ onto w , then ( )w w wu v u v+ = + :

u

u v+

u

v u v+ v

w w

wu wv

wu

wv

( )wu v+ ( )wu v+

So we can prove the distributive law for arbitrary vectors u , v , and w by considering the projections of u and v onto w : ( ) ( ) ( )w w wu v w u v w u v w+ ⋅ = + ⋅ = + ⋅ (1) Since wu , wv and w wu v+ are all parallel to w , ( )w w w wu v w u w v w+ ⋅ = ⋅ + ⋅ (2) Combining (1) and (2) we get:

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( ) ( )w w w wu v w u v w u w v w u w v w+ ⋅ = + ⋅ = ⋅ + ⋅ = ⋅ + ⋅ .

There are ways to define the multiplication of vectors with the product being a vector. One is defined for two-dimensional vectors, and it turns them into complex numbers. Complex numbers are the subject of Chapter <...>. The other product is called the cross product, u v× . The cross product is defined for three-dimensional vectors and is useful in physics. You can find the definition of the cross product on the Internet.

Exercises 1. Given four vectors of length 2 2 —

— find the dot product for each pair.

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2. In the next section we will discuss a simple formula for the dot product of two vectors with known coordinates. Given the dot product, find the angle between a given pair of vectors u and v : (a) (b) (c)

v

u

v

u

v

u

2u v⋅ = 12u v⋅ = 7u v⋅ = −

Hint: Find u and v by the Pythagorean theorem; use the 1cos− function on your calculator to find θ from cosθ .

3. h and m represent the hour and minute hands on a clock. 2h m⋅ = at noon,

3h m⋅ = at 1 pm, and 1h m⋅ = at 2 pm. Find h m⋅ on the hour for the rest of the day.

4. Prove the commutativity of the dot product. 5. Prove the associativity of the multiplication by a number and the dot product. 6. Prove the distributive law for the dot product for three vectors that are

parallel to the same line. 7. Show that vu v u v⋅ = ⋅ , where vu is the projection of u onto v . 8. If we defined u v⋅ simply as u v , which of the “nice” multiplication laws

would be broken? 9. Show that for any vectors u and v , 2 2 22u v u u v v+ = + ⋅ + . Hint:

Represent the squared lengths using dot products.

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10.

b a

C

B

A c

θ

In any triangle ABC with sides BC = a, AC = b and AB = c, and C θ∠ = ,

2 2 2 2 cosc a b ab θ= + − . This is known as the Law of Cosines. Prove this law using vectors. Hint: Superimpose vectors onto the sides of the triangle so that one of them is equal to the difference of the other two.

11. What is the distance between two points with polar coordinates 1 1( , )r θ and

2 2( , )r θ ? Hint: See Question 10. 12. Using dot products of vectors, show that the diagonals of a rhombus (a

parallelogram in which all sides are equal) are perpendicular to each other. 13. Using dot products of vectors, show that in any parallelogram the sum of the

squared lengths of the sides is equal to the sum of the squared lengths of the diagonals.

14. Using dot products of vectors, prove that a parallelogram in which diagonals

have the same length is a rectangle. 15.

Find the dot product of vectors drawn on the altitudes of an equilateral triangle with side 1.

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16.

A

N M

B C In an isosceles triangle ABC, AB = AC, M is the midpoint of AB , N is the midpoint of AC , and CM is perpendicular to BN . Determine the measure of A∠ . Hint: Consider AM , AC , AN , AB , and MC NB⋅ .

17. Show that for any positive integer N, 0

2cos 0N

k

kNπ

=

=∑ . Hint: See

Question <...> in Section <...>.

4.5 Numerical Representation Cartesian coordinates allow us to represent points on the plane as pairs of numbers. They also allow us to represent two-dimensional vectors as pairs of numbers. The easiest way to see that is simply to draw any vector from the origin:

0 1 2 3 4 -4 -3 -2 -1

-4 -3 -2 -1

1 2 3 4

(3.5, 2.25)

x

y

( )3.5, 2.25v =

This establishes a one-to-one correspondence between vectors and points on the plane: a vector corresponds to its endpoint. The zero vector (0, 0) corresponds to the origin. So we can represent any two-dimensional vector as a pair of numbers (x, y), which are the coordinates of the end of the vector when its beginning is at the origin.

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The same notation is used for point coordinates and for vectors: (x, y). We can decide from the context whether (x, y) represents a point or a vector.

x and y are called the components of the vector (x, y). If we draw a vector from an arbitrary point ( , )B BB x y on the coordinate plane, and the end of the vector is at the point ( , )E EE x y , then its components are

,E B E Bx x x y y y= − = − . For example:

O 1 2 3 4 -4 -3 -2 -1

-4 -3 -2 -1

1 2 3 4

( )3.5, 2.25v =

x

y

(1, 2)

(4.5, 4.25)

Suppose xv and yv are projections of a vector v onto the x- and y-axis, respectively. Then x yv v v= + . Let us consider unit vectors i and j , which have lengths equal

to 1 and point in the directions of the x- and y-axis, respectively. Then xv xi= ,

yv yj= , and x yv v v xi yj= + = + (Figure <...>). This is another very important way of looking at vector components: they are the coefficients when a vector is represented as a combination of unit vectors.

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O 1

( , )v x y=

x

y

j i xv

yv

1

Figure 4-8. ( , )v x y xi yj= = + .

By the Pythagorean theorem, 2 2 2v x y= + ; 2 2v x y= + .

The same principles apply to vectors in a three-dimensional space: ( , , )x y z xi yj zk= + + , where i , j , and k are unit vectors on the x-, y-, and z-axis. The same is true for n-dimensional vectors, if you can imagine that.

The beauty of the component representation of vectors is that vector arithmetic becomes very easy: given ( , )u uu x y= and ( , )v vv x y=

( , ) ( , ) ( , )u u v v u v u vu v x y x y x x y y+ = + = + +

( , ) ( , ) ( , )u u v v u v u vu v x y x y x x y y− = − = − −

( , ) ( , )k x y kx ky=

In other words, addition, subtraction, and multiplication by a number are performed on the corresponding individual components. We call this componentwise arithmetic.

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Example 1: (7.5, 2.3) + (1.1, 10) = (8.6, 12.3) (4.5, -2) + (-1, 6.2) = (3.5, 4.2) (4.5, -2) + (-1, 6.2) = (5.5, -8.2) (4.5, -2) - (-1, 6.2) = (5.5, -8.2) 2.5(3, 6) = (7.5, 15) (-2)(3, 5) = (-6, -10)

There is a neat formula for the dot product, too:

( , ) ( , )u u v v u v u vu v x y x y x x y y⋅ = ⋅ = +

Question <...> in the exercises asks you to show that this is consistent with the definition of the dot product given in Section <...>. This is also consistent with the formula for the length of a vector: 2 2 2( , ) ( , )v v v x y x y x y= ⋅ = ⋅ = + . The same is true for n-dimensional vectors:

1 2 1 2 1 1 2 2( , ,..., ) ( , ,..., ) ...n n n nu u u v v v u v u v u v⋅ = + + + and 2 2 2 21 2 ... nv v v v= + + + .

Example 2: (2, 1) ( 3, 5) 2( 3) 1 5 1⋅ − = − + ⋅ = −

Exercises 1. The coordinates of points A and B are (5, 7) and (3, 9) , respectively. Write

AB in the component form.

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2.

Represent each of the vectors shown above in component form. (One square on the grid is one unit.)

3. For each vector in Question 2, state its length. 4. If ( , )v x y= , what is v− ? 5.

(c)

v

u

(b)

v

u

(a)

v

u

(f)

v u

(e)

v u

(d)

v

u

For each given pair of vectors u and v above, represent each vector in the component form and state their sum and difference.

6. State (exactly) the lengths of the following vectors:

(a) (5, 12) (b) (1, 7) (c) (-3, 4) (d) (3, 4, 12) (e) (1, -2, 4, -8, 16, -32)

7. Find a vector that has the same direction as (3, -4) and whose length is equal

to 1. 8. Evaluate i i⋅ , j j⋅ , and i j⋅ .

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9. Show that ( , ) ( , )u u v v u v u vx y x y x x y y⋅ = + . Hint: ( , )x y xi yj= + ; use the distributive law.

10. Determine the angle, in radians, between the following vectors, using their

dot product.

(a) (3, 1) and (2, 5) (b) (3, 1) and (-2, 5) (c) (3, 1, 1) and (1, -2, 5)

11. What is the acute angle, in radians, between the lines 2y x= and 12

y x= ?

12. Find a vector that is perpendicular to (1, 5) and has the same length. How

many such vectors exist? 13. Give an example of n n-dimensional vectors (in the component form) such

that their lengths are all equal to 1, and any two of the vectors are perpendicular to each other.

14. The coordinates of the vertices of a triangle are (0, 4), (4, 5), and (5, 2).

Determine analytically (without using a sketch) whether this triangle is acute-angled, right, or obtuse-angled.

15. Prove the Schwarz inequality:

2 2 2 2 2 2 21 1 2 2 1 2 1 2( ... ) ( ... )( ... )n n n nu v u v u v u u u v v v+ + + ≤ + + + + + +

16. Suppose that 1x y z+ + = . When 13

x y z= = = , 2 2 2 13

x y z+ + = . Show

that 2 2 2 13

x y z+ + > in all other cases. Hint: See Question 15.

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17. 1x y z+ + = is the equation for the plane in three-dimensional space that contains the points (1, 0, 0), (0, 1, 0), and (0, 0, 1):

x

y

z

1 1

1 Find the distance from the origin to this plane. Hint: See Question 16.

4.6 Review Concepts, terms, methods, and formulas introduced in this chapter:

An equivalence relation and its properties: reflexivity, symmetry, transitivity Vectors: two-dimensional, three-dimensional, n-dimensional Geometric representation of a vector Numeric representation of a vector (the component form) Length (magnitude) and direction of a vector Addition: the parallelogram rule, the triangle rule Zero vector Multiplication by a number Vector space Dot product The component form: ( , )v x y xi yj= = +

θ p

v

O

V

P θ

p

v

O

V

P cos

pv

θ = − cospv

θ =

θ cos0 1=

cos 02π=

0 cos 1θ< < 1 cos 0θ− < <

cos 1π = −

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u

v u v+

O P

v

u

v

O

u v−

( , ) ( , ) ( , )u u v v u v u vx y x y x x y y+ = + +

( , ) ( , ) ( , )u u v v u v u vx y x y x x y y− = − −

( , ) ( , )k x y kx ky= cosu v u v θ⋅ = ⋅ ⋅

vu v u v⋅ = ⋅ , where vu is the projection of u onto v

2v v v⋅ = ; v v v= ⋅

0 0v⋅ =

0u v⋅ = when and u and v are perpendicular to each other

( , ) ( , )u u v v u v u vx y x y x x y y⋅ = +

1 2 1 2 1 1 2 2( , ,..., ) ( , ,..., ) ...n n n nu u u v v v u v u v u v⋅ = + + +

2 2 2 21 2 ... nv v v v= + + +

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Vectors - Skylight Publishing · to vectors. In physics and other applications, it is often...

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Transcript of Vectors - Skylight Publishing · to vectors. In physics and other applications, it is often...