VECTORS (Ch. 12)

Vectors in the plane

Definition: A vector v in the Cartesian plane is an ordered pair of real numbers: a,b .

We write v = a,b and call a and b the components of the vector v.

Geometric representation of vector v is the directed

line segment from origin O to point P(a,b).

A directed line segment has length and direction.OP

Length of a vector v = a,b is defined as

Example: The length of the vector v = 2,-3 is

The zero vector is 0= 0,0 has length zero and no direction.

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Algebraic operations

1. Two vectors v = v1, v2 and u = u1, u2 are equal if v1= u1 and v2 = u2.

2. The sum of two vectors v = v1, v2 and u = u1, u2: v + u = v1 + u1 , v2 + u2

3. If u = u1, u2 and c is a real number, then the scalar multiple cu is the vector

cu = cu1, cu2

Let a, b and c be vectors and r and s real numbers. Then

1. a + b = b + a

2. a + (b + c) = (a + b) + c

3. r(a + b) = r b + r a

4. (r + s) a = r a + s a

5. (rs) a = r(sa) = s(ra)

The unit vectors i and j- A unit vector is a vector of length 1.

- If a = a1, a2 0 then

- Special unit vectors: i = 1, 0 and j = 0, 1

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If a = a1, a2 , then

a = a1, 0 + 0, a2 = a1 1, 0 + a2 0, 1

= a1 i + a2 j.

So every vector in the plane is a linear combination of i and j.

Vectors in space (3- dimensional)

v = x, y , z with length:

can be written: v = x i + y j + z k, where

i = 1, 0 , 0 , j = 0, 1 , 0 , k = 0, 0 , 1

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Dot Product

Given two vectors

Theorem. If is the angle between a and b then the dot product is defined as

a b = |a| |b| cos .

cos = ( a b) / ( |a| |b| )

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The angle between a and b can be found using cos = ( a b) / ( |a| |b| ).

Properties: Let a and b be vectors and r a real number. Then a b = b a a a = | a|2

a ( b + c ) = a b + a c ( r a ) b = r (a b ) = a (r b)

CROSS PRODUCT

Length of the cross product

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Properties

a, b and c are vectors and r is a real number.

1. a x b = -(b x a)

2. a x (b x c) = (ac)b – (a b)c

3. a (b x c) = (a x b) c

4. a x ( b + c ) = (a x b) + (a x c)

5. ( r a ) x b = r (a x b ) = a x (r b)