Introduction to Vectors
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Transcript of Introduction to Vectors
Introduction to Vectors
UNIT 1
What is a Vector?
• A vector is a directed line segment, can be represented as AB or AB, where A and B are the two endpoints of the line segment.
• Directed means that the vector has a direction.
• QUESTION: Which direction is implied for vector AB?
Vectors: Example
A
B
Vector Quantity
• There are exactly two properties that completely characterize a vector:1) Direction – which way does the vector point?2) Magnitude – the length of the vector, written as |XY|
for vector XYTogether the direction and magnitude define the vector quantity.
• QUESTION: What are some examples of vectors that we are already familiar with?
Equal Vectors
• Two vectors are equal vectors if they have both the same direction and magnitude
Three cars on the road are driving in the same direction at the same speed:
Do they have equal velocity vectors?
Opposite Vectors: An opposite vector is a vector with the same magnitude as the original, but opposite direction:
A
B
In this illustration, AB and BA are opposite vectors.
Vector Sum
• A vector sum A + B is defined as a vector that results from placing the initial point of vector B at the terminal point of vector A: the vector with the same initial point as A and the same terminal point as B is the vector sum.
A
B
A + B
Parallelogram Rule
• Another method, called the parallelogram rule, to find A + B is to place vectors A and B so that their initial points coincide, then complete a parallelogram that has A and B as two adjacent sides. The diagonal of the parallelogram with the same initial points is the vector sum:
Parallelogram Rule Example
A
B
A + B
The two methods side by side:
A
B
A + BA
B
A + B
The two methods give identical results for the vector sum
QUESTION
• Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force:
22
15
100°
QUESTION
• Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force:
22
15=80
Z 22
100°
Scalar Product
• A real number k and a vector U create the vector kU which has the magnitude |k| times the magnitude of U . kU has the same direction as U if k>0, and the opposite direction if k < 0:– So vector 2U would be twice the length of U and
would point in the same direction as U does:
Scalar Product Examples
U
2U
3U
-U
-2U
-3U
Position Vectors
• A position vector is a vector with its initial point at the origin and with its endpoint at (a, b). It is written <a, b>, so U = <a, b> below:
(a, b)
U
X axis
Y axis
(0, 0)
Direction Angle of <a, b>
• The direction angle is the positive angle between the x-axis and a position vector:
(a, b)
U
X axis
Y axis
(0, 0)
Direction Angle
Direction Angle of <a, b> (cont)
(a, b)
U
X axis
Y axis
(0, 0)
The direction angle θ satisfies tan θ = b/a, where a ≠ 0:
b
a
Magnitude of vector <a, b>
• The magnitude of vector <a, b> is given by |U| =
(a, b)
U
X axis
Y axis
(0, 0)
b
a
QUESTION
• What is the direction angle and magnitude of vector U=<3, -2>?
(3, -2)U
X axis
Y axis
(0, 0)
QUESTION
• What is the direction angle and magnitude of vector <3, -2>?• ANSWER: Direction angle =
326.3°, magnitude =
Horizontal and Vertical Components of a Vector
• The horizontal and vertical components of a vector U are given by:a = |U|cos θ b = |U|sin θ
(a, b)
U
X axis
Y axis
(0, 0)
b = |U|sin θ
a = |U|cos θ
QUESTION
• Calculate the vertical and horizontal components of a vector with direction angle of 40° and a magnitude of 25.
(a, b)
|U|=25
X axis
Y axis
(0, 0)
= 40
QUESTION
• Calculate the horizontal and vertical components of a vector with direction angle of 40° and a magnitude of 25.• ANSWER: x = 19.2, y = 16.1 or
<19.2, 16.1>
Vector Operations Overview
• <a, b> + <c, d> = <a + c, b + d>• k*<a, b> = <ka, kb>• If A = <a1, a2>, then –A = <-a1, -a2>• <a, b> - <c, d> = <a, b> + -<c, d> OR <a, b> + <-c, -d> = <a – c, b – d>
Vector Operations
• For any real numbers a, b, c, and d:<a, b> + <c, d> = <a + c, b + d>
(a, b)
X axis
Y axis
(0, 0)(c, d)
(a+c, b+d)
Vector Operations
• For any real numbers a, b, c, and d:<a, b> + <c, d> = <a + c, b + d><3, 4> + <4, -2> = <7, 2>
(3, 4)
X axis
Y axis
(0, 0)(4, -2)
(7, 2)
Vector Operations (cont)
B
If U = <a1, a2>
Then –U = <-a1, -a2>
Vector Operations
• Scalar multiplication: k*<a,b> = <ka, kb>• Examples:
-3*<4, 7> = <-12, -21>6*<1, 2> = <6, 12>0*<3, 5> = <0, 0>
Vector Operations
(0, 0)
(3, 4)
(0, 0)
(6, 8)U
2U
QUESTION
• Consider the vectors shown in the following figure, and perform the operations:
UV
U + V(4, 3)
X axis
Y axis
(0, 0)
(-2, 1)
(x, y) a)U + Vb)-2Uc) 4U – 3V
Vector Subtraction
A
B
-A
-B
• Vector subtraction is the inverse operation of vector addition and is defined as adding the negative vector:So we have B – A = B + (-A) for all vectors A, BTherefore (see below) B – A = C
C
Vector Subtraction
• <a, b> - <c, d> = <a, b> + -<c, d> OR <a, b> + <-c, -d> = <a – c, b – d>• Examples:
<3, 6> - <4, 4> = <-1, 2><6, -4> - <-2, -5> = <8, 1><0, 1> - <4, 10> = <-4, -9>
Vector Subtraction QUESTIONS
• QUESTION: Express A as a difference of two vectors.
• QUESTION: Express B as a difference of two vectors.
A
B
C
Vector Notation Conventions
• Unit Vectors: i = <1, 0>, j=<0, 1>• i, j Form for Vectors: If v = <a, b>, then v = ai + bj
Unit Vectors
(5, 3)
U
X axis
Y axis
(0, 0) i ii ii
j
j
j
i j
U = 5i + 3j
QUESTION
• Write the vector in the form ai + bj:
<-5,8>
QUESTION
• Write the vector in the form ai + bj:
<-5,8>Answer: If v=<a,b>, then:V=ai + bj, so V = -5i + 8j
Dot Product
• The dot product of the two vectors U = <a, b> and V = <c, d> is denoted by U•V, read “U dot V,” is given by U•V = ac + bd.
• Examples: <2, 3>•<4, -1> = <6, 4>•<-2, 3> =
Geometric Interpretation of the Dot Product
• If θ is the angle between two nonzero vectors U and V, where 0° < θ < 180°, then
U•V = |U|*|V|cos θ• Example:
<2, 0>•<3, 3> = 6 using <a,b>•<c,d>=ac+bdBut it is also true using U•V = |U|*|V|cos θ
Geometric Interpretation of the Dot Product
(3, 3)
U
X axis
Y axis
(0, 0)
(2, 0)
V
U • V = |U||V|cos θ = (3
θ=45
Properties of the Dot Product
U•V = V•UU•(V+W)=U•V + U•W(U + V)•W = U•W + V•W(kU)•V = k(U•V) = U•(kV)0•U = 0U•U = |U|2
The Dot Product Can Be Positive, Zero, or Negative
θθ < 90: Positive dot product
θ
θ
θ > 90: Negative dot product
θ = 90: Zero dot product