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Chapter 6 Test
1. The vectors a ! , b !
, and c ! are shown.
a
b
c
Using these three vectors, demonstrate that a ! Qb ! c ! R Qa ! b ! R c ! .
Name this property and explain how your answer shows this to be true.
2. A12, 3, 52 and B16, 7, 32 are two points in R 3 . Determine each
of the following:
a. AB ! b. @AB ! @ c. a unit vector in the direction of BA !
3. The vectors and are each of length 3 units, i.e.,
If x ! determine
4. a. If 3x ! y ! x ! y !
and express the vectors and in terms
of a ! 2y
and b ! ! a ! 5x ! x !
3y ! y ! x ! y ! 3.
V17,
b ! .
,
x ! y !
.
b. Solve for a, b, and c: 12, 1, c2 1a, b, 12 312, a, 42 13, 1, 2c2.
5. a. Explain why the vectors a ! 12, 32 and b ! 13, 12 span R 2 .
b. Determine the values of p and q in p12, 32 q13, 12 113, 92.
6. a. Show that the vector can be written as a linear combination
of b ! a !
13, 1, 42 and c ! 11, 12, 292
11, 2, 32.
b. Determine whether can be written as a linear
combination of p ! r ! 116, 11, 242
12, 3, 42 and q ! 14, 1, 62. Explain the significance
of your result geometrically.
7. x ! and y ! are vectors of magnitude 1 and 2, respectively, with an angle of
between them. Determine 3x ! 2y ! and the direction of 3x ! 2y ! 120°
.
8. In triangle ABC, point D is the midpoint of and point E is the midpoint
of AC ! BC !
. Vectors are marked as shown. Use vectors to prove that DE ! 1 2 BA! .
D
a
B
a
C
b
E
b
A
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CHAPTER 6 TEST
NEL