INTRODUCTION TO VECTORS - PART 1
Example 1:
Find the component form of a vector with initial point P at (1, 2) and terminal point Q at (4, 4). Name the vector v.
Given and P and Q , we find the component form of the vector to be
< 4 1, 4 2 >, which equals v = < 3, 2 >
Please note that we moved the vector PQ to 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 Angle of 60o which is 60o to find that the direction angle must equal 300o.
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 Angle of 30o which is 30o to find that the direction angle must equal 210o.
Example 8:
Let v = < 2, 5 > and w = < 3, 4 >. Find v + w.
v + w = < 2, 5 > + < 3, 4 >
= < 2 + 3, 5 + 4>
= < 1, 9>
Example 9:
Let v = < 2, 5 > and w = < 3, 4 >. Find the following:
(a) v
v = (1)v
= (1)< 2, 5 >
= < (1)(2), (1)(5) >
= <2, 5>
(b) 2v
2v = 2< 2, 5 >
= < 4, 10 >
(c) w v
w v = w + (1)v (Adding the negative of vector v!)
= < 3, 4 > + (1)< 2, 5 >
= < 3, 4 > + < 2, 5 >
= < 5, 1 >
(d) v + 2w
v + 2w = < 2, 5 > + 2< 3, 4 >
= < 4, 13 >