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10. Parallelogram is determined by the vectors and
OB ! OBCA
OA ! 16, 32
111, 62.
a. Determine OC !
, , and .
b. Verify that @ OA ! BA !
@ @ BC ! BC !
@ .
11. ^ABC has vertices at A12, 32, B16, 62, and C14, 112.
a. Sketch and label each of the points on a graph.
b. Calculate each of the lengths @ AB ! @ , @ AC ! @ , and @ CB ! @ .
c. Verify that triangle ABC is a right triangle.
12. A parallelogram has three of its vertices at A11, 22, B17, 22, and C12, 82.
a. Draw a grid and locate each of these points.
b. On your grid, draw the three locations for a fourth point that would make
a parallelogram with points A, B, and C.
c. Determine all possible coordinates for the point described in part b.
13. Determine the value of x and y in each of the following:
a. 31x, 12 512, 3y2 111, 332
b. 21x, x y2 316, y2 16, 42
C
T
14. Rectangle ABCD has vertices at A12, 32, B16, 92, C1x, y2, and D18, 112.
a. Draw a sketch of the points A, B, and D, and locate point C on your graph.
b. Explain how you can determine the coordinates of point C.
15. A15, 02 and B10, 22 are points on the x- and y-axes, respectively.
a. Find the coordinates of point P1a, 02 on the x-axis such that .
b. Find the coordinates of a point on the y-axis such that @ QB ! @ PA ! @
@ @ QA ! @ PB ! @
@ .
PART C
16. Find the components of the unit vector in the direction opposite to , where
OP ! and OQ ! PQ !
111, 192 12, 212.
17. Parallelogram OPQR is such that OP ! 17, 242 and .
a. Determine the angle between the vectors OR ! OR !
and OP ! 18, 12
.
b. Determine the acute angle between the diagonals OQ !
and RP !
.
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6.6 OPERATIONS WITH ALGEBRAIC VECTORS IN R 2
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