1. This document provides a study guide for precalculus chapter 8 covering vectors and parametric equations. 2. It includes vocabulary terms, examples of finding vector magnitudes and directions, writing vectors in standard form, vector operations including addition, subtraction and inner/outer/cross products, writing parametric equations of lines, and word problems involving vector and parametric representations of motion. 3. The study guide provides a review of key concepts and skills along with practice problems to help prepare for an assessment on chapter 8 material involving vectors and parametric equations.

1. Precalculus
Chapter 8 Study Guide
Vocabulary:
1. The _____ of two or more vectors is vector.
the sum of the vectors.
2. A _____ vector has a magnitude of components
one. cross
3. Two vectors are equal if and only if direction
they have the same direction and ____. equal
4. The ____ product of two vectors is a inner
vector. magnitude
5. Two vectors in space are perpendicular parallel
if and only if their ____ product is zero. parameter
6. Vectors with the same direction and resultant
different magnitudes are ____. scalar
7. A vector with its initial point at the standard
origin is in ____ position. unit
8. Two or more vectors whose sum is a vector
given vector are called ____ of the given
Find the magnitude and direction of the following vectors:
r r r
1. s 2. t 3. u
r ρ r ρ ρ r ρ
4. 3u − τ 5. s +τ−υ 6. s − 2τ
Write the vectors as an ordered pair or triple (i.e., standard form). Then find the
magnitude.
æ - 3ö æ 5ö
7. (9, 4) and (4,16) 8. ç −9, ÷ and ç - 6, ÷
è 2ø è 2ø
9. ( 2, 3, −4 ) ανδ ( 2, 3, −4 ) 10. ( 2, 3,−4 ) ανδ (−2, −3, 4)
r r r
Find an ordered pair to represent a in each equation if b = 6, 3 and c = −4,8 .
r ρ ρ r 1 ρ ρ r ρ ρ ρ
11. a = −β + 4 χ 12. a = ( 2β − 5 χ) 13. a = ( 3β + χ) + 5β
3

2. r r u
r
Find an ordered triple to represent u in each equation if v = 4, −3,5 , w = 2, 6, −1 , and
r
z = 3, 0, 4 .
r 1ρ υ ρ ρ r υ ρ
ρ r ρ 2υ ρ ρ
14. u = ϖ− ω + 2 ζ 15. u = −4 ω + ζ 16. u = 3ϖ− ω + 2 ζ
2 3
17. Write the vectors from 7 – 16 as a sum of unit vectors.
Find the inner product. State whether the vectors are perpendicular.
18. 4, 8 ⋅ 6, −3 19. 5, −1 ⋅ −3, 6 20. 8, 4 ⋅ 2, 4
21. 3,1, 4 ⋅ 2,8, −2 22. −2, 4,8 × 16, 4, 2 23. 7,−2, 4 ⋅ 3, 8,1
Find the cross product.
24. 0,1, 2 × 1,1, 4 25. 3, 2,0 × 1, 4, 0 26. −3,- 1, 2 ´ 4, - 4, 0
27. Find a vector perpendicular to the plan containing the points (1,2, 3) , ( −4, 2, - 1) ,
and ( 5, −3, 0 ) .
r ρ ρ ρ ρ ρ ρ
28. Show that a × ( β + χ) = ( α × β) + ( α × χ) .
u ρ ρ υ
r ρ
29. Explain whether the equation m × ν = ν × µ is true.
r
Write a vector equation of the line that passes through point P and is parallel to a . Then
write the parametric equations of the line.
ρ ρ ρ
30. P ( 5, 7 ), α = 2, 0 31. P ( −1, 4 ), α = 6,−10 32. P (1, 5 ), α = −7,2
Write parametric equations of each line with the given equation.
33. −3x + 4y = 7 34. 9x + ψ= −1 35. −4x + y = - 2
Write an equation in slope-intercept form of the line with the given parametric equations.
x = 2τ x = 4 τ − 11 x = 3 + 2τ
36. 37. 38.
ψ= 1 − τ ψ= τ + 3 ψ= −1 + 5τ
39. Graph the parametric equations x = χοσ τ ανδ ψ= σ τ .
2
ιν 2
40. Astronomers have traced the path of two asteroids traveling through space. At a
particular time t, the position of asteroid Ceres can be represented by
( −1 + t, 4 - t, - 1 + 2t ) . Asteroid Pallas’ path at any time can be expressed by
( −7 + 2t, - 6 + 2t, - 1 + t ) .
a) Write the parametric equations for the path of each asteroid.
b) Do the paths cross? If so, where?
41. Find the parametric equations for the line passing through points
æ 1 ö
ç − ,1,1÷ and ( 0, 5,- 8 ) . [Note: Equations needed for x, y, and z]
è 3 ø
42. Find the initial horizontal and vertical velocities of a soccer ball kicked with an

3. initial velocity of 33 feet per second at an angle of 29° with the ground.
43. A rock is tossed at an initial velocity of 50 meters per second at an angle of 8°
with the ground. After 0.8 seconds how far has the rock traveled horizontally and
vertically?
44. a) Write a matrix for the vertices of the given
prism.
b) Translate the figure using the vector
r
n = 2, 0, 3 .
c) Transform the original figure using the
 1 0 0 
 
matrix M =  0 −1 0  .
 0 0 1 
 