VectorsTS: Explicitly assessing information and drawing conclusions.

Warm Up:a) What are the coordinates of A and B b) What is the distance between A and B?

Vectors are directed line segments that are typically used to represent a force or speed (things that have both a magnitude and direction). The magnitude is indicated by the

length of the vector, and the direction is indicated by the slope

(or angle) of vector.

If a plane is flying due North East at 100mph, then it’s motion can be represented as a vector. Which is shown as

a terminating ray (or “directed segment”)

N100mph

45°

If a person is pushing a box up a hill which has an angle of elevation of 15° using 16 Newtons (N) of force to do it, then

it’s force can be shown using a vector, shown as this terminating ray (or “directed segment”)

16N

15°

Equivalent Directed Line Segments

BA

C

Component form of a Vector

The component form of a vector with initial point P (p1, p2) and terminal point Q (q1, q2) is given by

The magnitude of v is given by

SPECIAL VECTORS:

If ||v|| = 1, v is a unit vector.

If ||v|| = 0, v is the zero vector.

1 1 2 2 1 2, ,PQ q p q p v v v��������������

2 2 2 21 1 2 2 1 2( ) ( )q p q p v v v

v1

v2

||v||

P

Q

Ex: Find the component form and magnitude of the vector shown, then drawn an equal

vector whose terminal point is (5, 2)

Vector Operations: Scalar Multiplication

Given v find:

1) 2v

2) -v

3) 0.5v

1 2 1 2, ,k k u u ku ku u

Vector Operations: Addition

Given u and v

Find u + v

1 1 2 2,u v u v u v

u

v

Vector Operations: Subtraction

Find u - v

u

v

Example: Multiple Operations

Find u – 2v

u

v

Properties of Vector Addition and Scalar Multiplication

Let u, v, and w be vectors and let c and d be scalars. Then the following properties are true.

1) u + v = v + u

2) u + 0 = u

3) c(du) = (cd)u

4) c(u + v) = cu + cv

5) ||cv|| = |c| ||v||

6) (u + v) + w = u + (v + w)

7) u + (-u) = 0

8) (c + d)u = cu + du

9) 1(u) = u

10) 0(u) = 0

Unit Vectors

To find a unit vector divide the vector, v, by its magnitude.

This will have the same direction as the vector, v, but it’s magnitude (or length) will be 1.

unit vector v

v

Example:

Find a unit vector going in the same direction as < -2, 5 >. Verify it is a unit vector.

Writing a vector as a Linear Combination of Unit Vectors, &

Unit Vectors: = <1, 0> and = <0, 1>

Linear Combination of Unit Vectors

v = < v1, v2> = v1 + v2

i j

i j

i j

Example:

If u is a vector with initial side (2, -5) and terminal side (-1, 3), write u as a linear combination of the standard unit vectors and i j