Determining the components of a vector perpendicular to two nonzero vectors
will prove to be important in later applications.
IN SUMMARY
Key Idea
• The dot product is defined as follows for algebraic vectors in R2 and R3,
respectively:
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! !
• If a 1a1, a2 2 and b 1b1, b2,2 , then a # b a1b1 a2b2
!
!
! !
• If a 1a1, a2, a3 2 and b 1b1, b2, b3 2 , then a # b a1b1 a2b2 a3b3
Need to Know
• The properties of the dot product hold for both geometric and algebraic
vectors.
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• Two nonzero vectors, a and b , are perpendicular if a # b 0.
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• For two nonzero vectors a and b , where u is the angle between the
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a#b
vectors, cos u ! ! .
0a 0 @ b @
Exercise 7.4
PART A
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1. How many vectors are perpendicular to a 11, 1 2? State the components
of three such vectors.
2. For each of the following pairs of vectors, calculate the dot product and, on
the basis of your result, say whether the angle between the two vectors is
acute, obtuse, or 90°.
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a. a 12, 12, b 11, 2 2
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b. a 12, 3, 12 , b 14, 3, 17 2
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c. a 11, 2, 52 , b 13, 2, 22
3. Give the components of a vector that is perpendicular to each of the following
planes:
a. xy-plane
b. xz-plane
c. yz-plane
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385
4. a. From the set of vectors e 11, 2, 1 2, 14, 5, 6 2 , 14, 3, 10 2, Q5, 3, 6 R f ,
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select two pairs of vectors that are perpendicular to each other.
b. Are any of these vectors collinear? Explain.
5. In Example 5, a vector was found that was perpendicular to two nonzero vectors.
a. Explain why it would not be possible
to do this in R2 if we selected the
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two vectors a 11, 22 and b 11, 1 2.
b. Explain, in general, why it is not possible to do this if we select any two
vectors in R2.
K
PART B
6. Determine the angle, to the nearest degree, between each of the following
pairs of vectors:
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a. a 15, 32 and b 11,2 2
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b. a 11, 42 and b 16, 22
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c. a 12, 2, 12 and b 12, 1, 2 2
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d. a 12, 3, 62 and b 15, 0, 12 2
7. Determine k, given two vectors and the angle between them.
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a. a 11, 2, 32 , b 16k, 1, k2 , u 90°
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b. a 11, 12 , b 10, k2 , u 45°
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8. In R2, a square is determined by the vectors i and j .
a. Sketch the square.
b. Determine vector components for the two diagonals.
c. Verify that the angle between the diagonals is 90°.
C
9. Determine the angle, to the nearest degree, between each pair of vectors.
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a. a 11 V2, V2, 12 and b 11, 1 2
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b. a 1 V2 1, V2 1, V2 2 and b 11, 1, 12
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10. a. For the vectors a 12, p, 8 2 and b 1q, 4, 122 , determine values of p and
q so that the vectors are
i. collinear
ii. perpendicular
b. Are the values of p and q unique? Explain why or why not.
11. ^ ABC has vertices at A12, 5 2 , B14, 11 2 , and C11, 6 2 . Determine the angles
in this triangle.
386 7 . 4 T H E D OT P R O D U C T O F A L G E B R A I C V E C TO R S
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z
12. A rectangular box measuring 4 by 5 by 7 is shown in the diagram at the left.
D
a. Determine the coordinates of each of the missing vertices.
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b. Determine the angle, to the nearest degree, between AE and BF .
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13. a. Given the vectors p 11, 3, 0 2 and q 11, 5, 2 2 , determine the
components of a vector perpendicular to each of these vectors.
y
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b. Given the vectors m 11, 3, 4 2 and n 11, 2, 32 , determine the
components of a vector perpendicular to each of these vectors.
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14. Find the value of p if the vectors r 1 p, p, 12 and s 1 p, 2, 3 2 are
perpendicular to each other.
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15. a. Determine
the algebraic condition such that the vectors c 13, p, 1 2
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and d 11, 4, q 2 are perpendicular to each other.
E
F
P( 7, 4, 5)
C
O
A
B
x
A
b. If q 3, what is the corresponding value of p?
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16. Given the vectors r 11, 2, 1 2 and s 12, 4, 22 , determine the
components of two vectors perpendicular to each of these vectors. Explain
your answer.
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4
17. The vectors x 14, p, 2 2 and y 12, 3, 6 2 are such that cos1Q21R u,
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where u is the angle between x and y . Determine the value(s) of p.
PART C
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18. The diagonals
of a parallelogram are determined by the vectors a 13, 3, 0 2
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and b 11, 1, 2 2 .
a. Show that this parallelogram is a rhombus.
b. Determine vectors representing its sides and then determine the length of
these sides.
c. Determine the angles in this rhombus.
T
19. The rectangle ABCD has vertices at A11, 2, 3 2 , B12, 6, 9 2 , and D13, q, 8 2 .
a. Determine the coordinates of the vertex C.
b. Determine the angle between the two diagonals of this rectangle.
20. A cube measures 1 by 1 by 1. A line is drawn from one vertex to a diagonally
opposite vertex through the centre of the cube. This is called a body diagonal
for the cube. Determine the angles between the body diagonals of the cube.
a
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