12

12.2 VECTORS

DEFINITIONS:

Vectors, Initial and Terminal Point, Length

A vector is a directed line segment.

.B

terminal point

A.

Initial point

The directed line segment has initial point A and terminal point B; its length is denoted by . Two vectors are equal if they have the same length and direction.

Vector in the standard form

DEFINITION Component Form

If v is a two-dimensional vector in the plane equal to the vector with initial point at the origin and terminal point , then the component form of v is

v =

If v is a three-dimensional vector equal to the vector with initial point at the origin and terminal point , then the component form of v is

v =

or represent the magnitude or length of v.The magnitude or length of the vector is the nonnegative number

Example:1

Find the component form and the magnitude (length) of the vector where and

Example:2

A boy pulled a trolley along an inclined plane of 30o with a force of 10 lb making an angle of 45o with the horizontal. What is the effective force moving the trolley upward?

Vector Algebra Operation

Definitions: Vector Addition and Multiplication of a vector by a scalar

Let u = and v = be vectors with k a scalar.

Addition u + v =

Scalar Multiplication k u =

Illustration of Parallelogram Law of Addition and Resultant Vector.

The difference u v ; The unit vector, i, j, kExample:3

Given u = and v = . Find

(a) 5u -2v

(b) 4v + 3u

(c) A vector twice the magnitude of u and acts in the opposite direction of v.

(d) A vector of magnitude 5 and acts in the direction of u.Properties of vector OperationsLet u,v,w be vectors and a, b are scalars

1. u + v = v + u2. (u + v) + w = u + (v + w)

3. u + 0 = u4. u + (-u) = 05. 0u = 06. 1u = u7. a (b u) = (ab) u

8.a (u + v) = a u + a v

9. (a + b) u = a u + b u

Midpoint of a line Segment

Midpoint of a line segment joining points and is the point

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