Math 150 – Fall 2015
Chapter 8 & 9 Review
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Chapter 8 & 9 Review
Formulas: For the new material (chapters 8 & 9) these are the formulas you need to
MEMORIZE:
π
• sin−1 x: Domain [−1, 1], Range [ −π
2 , 2]
• cos−1 x : Domain [−1, 1], Range [0, π]
π
• tan−1 x. Domain (−∞, ∞), Range ( −π
2 , 2)
• Law of Sines:
sin A
a
=
sin B
b
=
sin C
c
• Law of Cosines: c2 = a2 + b2 − 2ab cos C
• 2-dimensions: ~i = h1, 0i, ~j = h0, 1i
• 3-dimensions: ~i = h1, 0, 0i, ~j = h0, 1, 0i, ~k = h0, 0, 1i
p
• Vector length ||hx1 , x2 , . . . , xn i|| = x21 + x22 + · · · + x2n
• Dot Product: hx1 , x2 , . . . , xn i · hy1 , y2 , . . . , yn i = x1 y1 + x2 y2 + · · · xn yn
• Angle between vectors:
~x · ~y = ||x|| ||y|| cos θ
or
cos θ =
~x · ~y
||~x|| ||~y ||
Note. These are the formulas you need to memorize for Chapter 8 & 9, but you also
need to know how to solve the problems!
Example 1. Find a unit vector which points in the same direction of the vector from
the point P (−3, 5) to the point Q(0, −1).
Example 2. Determine the angle between the vectors h−2, 4i and h3, 6i.
Example 3. Suppose a man walks 100 yards in a direction 30◦ north of east. Assuming
that the origin is the man’s starting point and that east is the positive x-axis, what are
the coordinates of the man’s final location?
Example 4. Find the length of the vector from the point P (2, 1, −5) to the point
Q(0, −3, 2).