Algebra and Trigonometry, 2015a

SECTION 10.8 VECTORS 849
y
5 (4, 5)
4
(−3, 2) 3
(7, 3)
2
(0, 0) 1 Position vector
x
–4 –3 –2 –1 1 2 3 4 5 6 7 8
–1
–2
–3
–4
–5
Try It #1
Figure 4
Draw a vector v that connects from the origin to the point (3, 5).
Finding Magnitude and Direction
To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the
Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.
magnitude and direction of a vector
Given a position vector v = 〈a, b〉, the magnitude is found by |v | = √ — a 2 + b 2 . The direction is equal to the angle
formed with the x-axis, or with the y-axis, depending on the application. For a position vector, the direction is
found by tan θ = ( b __ a ) ⇒ θ = tan −1 ( b __ a ) , as illustrated in Figure 5.
y
5
4
3
2
1
– 1 – 1
〈a, b〉
|v|
θ
1 2 3 4 5
x
Figure 5
Two vectors v and u are considered equal if they have the same magnitude and the same direction. Additionally, if
both vectors have the same position vector, they are equal.
Example 3 Finding the Magnitude and Direction of a Vector
Find the magnitude and direction of the vector with initial point P(−8, 1) and terminal point Q(−2, −5). Draw the vector.
Solution First, find the position vector.
u = 〈−2, − (−8), −5−1〉
= 〈6, − 6〉
We use the Pythagorean Theorem to find the magnitude.
|u | = √ — (6) 2 + (−6) 2
= √ — 72
= 6 √ — 2