132 STEP 4. Review the Knowledge You Need to Score High
Example 1
Find the magnitude and direction of the vector represented by〈6,− 3 〉.
Step 1: Calculate the magnitude‖r‖=√
(6)^2 +(−3)^2 =√
45 = 3√
5.
Step 2: The terminal point of the vector is in the fourth quadrant. Calculate θ =
tan−^1(
− 3
6)
=tan−^1(
− 1
2)
≈−.464 radians. This angle falls in quadrant IV.Example 2
Find the magnitude and direction of the vector represented by〈−5,− 5 〉.
Step 1: Calculate the magnitude‖r‖=√
(−5)^2 +(−5)^2 =√
50 = 5√
2.
Step 2: The terminal point of the vector is in the third quadrant. Calculate
tan−^1(
− 5
− 5)
=tan−^1 (1)=
π
4
radians. The direction angle isθ=
π
4
+π=
5 π
4.
Example 3
Find the magnitude and direction of the vector represented by〈−1,√
3 〉.Step 1: Calculate the magnitude‖r‖=√
(−1)^2 +(√
3)^2 =√
4 =2.
Step 2: The terminal point of the vector is in the second quadrant. Calculatetan−^1(√
3
− 1)
=tan−^1 (−√
3)=−
π
3
radians. The direction angle is θ=−
π
3+
π=
2 π
3.
Example 4
Find the ordered pair representation of a vector of magnitude 12 and direction
−π
4
.x=12 cos(
−π
4)
= 6√
2 andy=12 sin(
−π
4)
=− 6√
2 so the vector is〈
6√
2,− 6√
2〉
.Vector Arithmetic
IfCis a constant,r 1 =〈x 1 ,y 1 〉andr 2 =〈x 2 ,y 2 〉, then:
Addition:r 1 +r 2 =〈x 1 +x 2 ,y 1 +y 2 〉
Subtraction:r 1 −r 2 =〈x 1 −x 2 ,y 1 −y 2 〉
Scalar Multiplication:Cr 1 =〈Cx 1 ,Cy 1 〉
Note:‖Cr 1 ‖=‖C‖·‖r 1 ‖
Dot Product: The dot product of two vectors isr 1 ·r 2 =‖r 1 ‖·‖r 2 ‖·cosθ
orr 1 ·r 2 =x 1 x 2 +y 1 y 2.