# Algebra and Discrete Grade 12

```MCV 4U
1. Given
 
a) y  z
Unit 2: Algebraic Vectors Extra Review
Solutions
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x  2iˆ  ˆj  5kˆ,
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y   1,2,2, z  3,2,4
A(1,3,4), B(3,2,7)
Find:
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b) z  x
c) BA
d) BA
e) ŷ
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f) the angle between x and
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g) 2 x  3 y  z
 
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h) 2 x  y   3x  2 y 


 
i) 3x  2 y   2 x  y 

z
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2. Draw the position vector u  2,3,5 . Be sure to draw the x, y, and z axis and label them.
3. Using vectors demonstrate that the following points are collinear. A(-1,3,-7),
B(-3,4,2) and C(5,0,-34)
4. A 60 N force is applied to the end of a 30 cm wrench and makes an angle of 40 with the handle of
the wrench. What is the magnitude of the torque on the bolt at the other end of the wrench.
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5. Find the work done if a force of 35 N in the direction of r  4,5,8 moves an object from G(1,7,-3)
to H(5,3,-2).
44
41.6
41.6
150.4 J
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6. A) If x  3t ,5,1 and y  t  2, t  3,5 are perpendicular, find all value(s) for t.
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B) For x  14,4k ,7 and y  2c,8,21 , find values for k and c if x is parallel with y .
7. Given W(-1,4,2), X(6,-2,3) and Y(-3,5,1), find:
a) area of triangle WXY
b) the co-ordinates of the point Z if WXYZ is a parallelogram.
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8. Given u  3,1,4 and v   2,4,1 , find the Pr oj u onto v  as an algebraic vector.
9. If OA, OB, and OC are three edges of a parallelepiped where O is (0,0,0), A is
(2,4,-3), B is (4,6,2), and C is (5,0,-2), find the coordinates of the other vertices of the parallelepiped
and find the volume of the parallelepiped.
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V ppp'd  a  b  c
V ppp'd  ( 26,  16,  4)  (5, 0,  2
V ppp'd  130  8
V ppp'd  138 u 3
12. Given :
and
a  5 and b  8
Find:
a) a  b
{29.73 ê in}
b) the area of the parallelogram formed by a and b .
{29.73 units squared}
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13. Given vectors b  (2,1,7) and c  (5,2,1) . Find c  b
{(-15,-37,-1)}
14. Given A(2,2,5 ) , B(2, 15
, ) and C( 6 , 1, 5 ) , find the area of triangle ABC. {25.7 u squared}
15. Determine the magnitude of the moment of the force about the fixed point A for each of the following:
a)
b)
{704.2 Joules}
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16. Find a vector  to both a = (1,1,2) and b = (2,1,- 2).
{769.1 Joules}
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17. Given: a  (1,2,2) and b  (2,1,3)
find each of the following:
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a. the (vector) projection of a on b
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c. any vector  to a

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d. a unit vector  to both a and b .
e. the area of a trianglewith
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f. the angle between a and b

 

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18. Given that d  5 , c  8 and the angle between d and c is 36  . Find 3d  c  2d  c { 117.6 ê }
19. Expand and simplify:
 


a) 2 x  y   3x  2 y 

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 
b) 3x  2 y   2 x  y 
```