Algebraic Vectors

Section 8-2

WHAT YOU WILL LEARN:

1. How to find ordered pairs that represent vectors.

2. How to add, subtract, multiply and find the magnitude of vectors algebraically.

Representing Vectors with Ordered Pairs

Bits of Information -As we found in the last homework, calculating resultants and doing “vector” math using drawing can be extremely inaccurate.

Vectors can be represented using ordered pairs.

The vector <3, 5> can represent a vector in standard position with its terminal point at (3, 5). This would be “like” a resultant of a horizontal vector of 3 and a vertical vector of 5.

Lastly, a vector does not actually have to be in standard position (at the origin) for us to be able to use it in algebraic expressions.

Magnitude of a Vector

212

21221 )()( yyxxPP

),(),( 121221 yyxxPP

You can find the magnitude of a vector that is represented by a set of ordered pairs using the following:

Given P1(x1, y1) and P2(x2, y2):

The ordered pair that represents a vector is found using:

Example

x-10 -5 5 10

y

-10

-5

5

10

• Write the ordered pair that represents the vector from

X(-3, 5) to Y(4, -2). Then find the magnitude of XY.

You Try• Write the ordered pair that represents the vector

from C(7, -3) to D(-2, -1). Then find the magnitude of CD.

Vector Operations

• Vectors that are represented by ordered pairs can be added, subtracted and multiplied by a scalar.

Addition

a + b = <a1,a2>+<b1,b2> = <a1+ b1, a2+ b2>

Subtraction

a - b = <a1,a2>-<b1,b2> = <a1 - b1, a2 - b2>

Scalar Multiplication

ka = k <a1,a2> = <ka1,ka2>

Example• Let m = <5, 7>, n = <0, 4>, and p=<-1,3> Find

each of the following:

A. m + p B. m – p

C. 7p D. 2m + 3n - p

You Try• Let u = <1, 4> and v = <0, 8>. Find each of the

following:

A. u + v B. u – v

C. ½ v D. 2u + 3v

An Application• Two paramedics are moving a person on a

stretcher. One person is pushing the stretcher with a force of 135 newtons at 58o with the horizontal, while the other person was pulling the stretcher with a force of 214 newtons at 43o with the horizontal. What is the magnitude of the force exerted on the stretcher?

You Try• Two people are pushing a patient on an MRI

machine cot. One person is pushing the cot with a force of 120 newtons at 55o with the horizontal, while the other person is pulling the cot with a force of 200 newtons at 40o with the horizontal. What is the magnitude of the force exerted on the cot?

Some Last Bits

A vector that has a magnitude of one unit is called a unit vector. A unit vector in the direction of the positive x-axis is represented by i, and a unit vector in the direction of the positive y-axis is represented by j. So, i = <1, 0> and j = <0, 1>.

Any vector a = <a1, a2> can be expressed as a1i + a2j

Example

Write AB as the sum of unit vectors for A(4,-1) and B(6, 2).

Step 1: Write AB as an ordered pair:

Step 2: Then write AB as the sum of unit vectors:

You Try

• Write AB as the sum of unit vectors for A(2, -7) and B(-1, 5).

Homework

page 497, 14-28 even, 44

page 497, 36-40 even