** 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 new 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 >I like to remember the formula as "terminal minus initial". We do it for the q's and the p's!

Please note that we moved the vector

PQto standard position without changing its magnitude and direction.

Example 2:

Find the component form of the vector with initial point

atPand terminal point(1, 3)atQ.(6, 5)Given vector

v= and pointsandP, we can calculateQ

v=<6 (1), 5 (3) >, which equals v = < 5, 8 >.I like to remember the formula as "terminal minus initial". We do it for the q's and the p's!

Example 3:

Calculate the EXACT magnitude of

v= < 3, 2 >.

Given ||

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

Example 4:

Calculate the EXACT magnitude of

v=< 0, 7 >.

Given ||

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

Example 5:

Calculate the EXACT magnitude of

v=< 4, 3 >.

Given ||

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

Example 6:

Calculate the EXACT magnitude of

v=< 8, 0 >.Given ||

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

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!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 8:

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 9:

Let

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

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

= < 2 + 3, 5 + 4>

= < 1, 9>

Example 10:

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 >