** INTRODUCTION TO VECTORS - PART 1**

Example 1:

Find the component form of a vector with initial point

atPand terminal point(1, 2)atQ. Name the vector(4, 4)v.Given and

andP, we find the component form of the vector to beQ< 4 1, 4 2 >

,which equalsv= < 3, 2 >

Please note that we moved the vector

PQto standard position without changing its magnitude and direction.

Example 2:

Calculate the EXACT magnitude of

v= < 3, 2 >.

Given ||

v|| = , we find ||v|| =

Example 3:

Calculate the EXACT magnitude of

v=< 0, 7 >.

Given ||

v|| = , we find ||v|| =

Example 4:

Calculate the EXACT magnitude of

v=< 4, 3 >.

Given ||

v|| = , we find ||v|| =

Example 5:

Calculate the EXACT magnitude of

v=< 8, 0 >.Given ||

v|| = , we find ||v|| =

Example 6:

Find the direction angle of vector .

Let's look at a picture of the vector and its direction angle.

Remember that a direction angle is positive and measured from the positive x-axis to the vector in standard position as shown below!From earlier work we know that , therefore, we find

and

Looking at the graph of the vector, we know that we are required to find a positive angle whose terminal side is in Quadrant IV.

We will use the

Reference Angleof60which is^{o}60^{o}to find that the direction angle must equal300.^{o}

Example 7:

Find the direction angle of vector .

Let's look at a picture of the vector and its direction angle.

Remember that a direction angle is positive and measured from the positive x-axis to the vector in standard position as shown below!Knowing that , we can find

and

Looking at the graph of the vector, we know that we are required to find a positive angle whose terminal side is in Quadrant IV. We will use the

Reference Angleof30which is^{o}30to find that the direction angle must equal^{o}210.^{o}

Example 8:

Let

v= < 2, 5 > andw= < 3, 4 >. Findv+w.

** v** + **w ***=* < 2, 5 > + < 3, 4 >

= < 2 + 3, 5 + 4>

= < 1, 9>

Example 9:

Let

v= < 2, 5 > andw= < 3, 4 >. Find the following:(a)

v

v=(1)v

=(1)< 2, 5 >

=< (1)(2), (1)(5) >= <2, 5>

(b) 2

v

2v= 2< 2, 5 >= < 4, 10 >

(c)

wv

wv=w+ (1)v(Adding the negative of vectorv!)

=< 3, 4 >+(1)< 2, 5 >

=< 3, 4 > + < 2, 5 >= < 5, 1 >

(d)

v+ 2w

v+ 2w=< 2, 5 > + 2< 3, 4 >= < 4, 13 >