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8. The vectors a ! , b ! , and c ! are as shown.
a. Construct the vector a ! b ! c ! b
.
a
b. If the vectors and are perpendicular, and if and
determine @ a ! a ! b ! b !
a ! 4
@ .
c
@ b ! @ 3,
9. Given p ! 111, 72, q ! 13, 12, and r ! 11, 22, express each vector
as a linear combination of the other two.
10. a. Find an equation to describe the set of points equidistant from A12, 1, 32
and B11, 2, 32.
b. Find the coordinates of two points that are equidistant from A and B.
11. Calculate the values of a, b, and c in each of the following:
a. 21a, b, 42 1 16, 8, c2 317, c, 42 124, 3, 252
2
b. 2 aa, a, 1 2 ab 13b, 0, 5c2 2 ac, 3 c, 0b 13, 22, 542
2
12. a. Determine whether the points A11, 1, 12, B12, 2, 22, and C14, 2, 12
represent the vertices of a right triangle.
b. Determine whether the points P11, 2, 32, Q12, 4, 62, and R11, 2, 32
are collinear.
13. a. Show that the points A13, 0, 42, B11, 2, 52, and C12, 1, 32 represent
the vertices of a right triangle.
b. Determine cos ABC.
14. In the following rectangle, vectors are indicated by the direction
of the arrows.
A
E
B
D
C
a. Name two pairs of vectors that are opposites.
b. Name two pairs of identical vectors.
c. Explain why @ AD ! @
2 @ DC
! @
2 @ DB
! @
2 .
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