VECTORS
135
BOARD PROBLES EXERCISE – II Q.1 Q.2 Q.3
Q.4 Q.5
Find if a 4ˆi ˆj kˆ and b ˆi 2ˆj 2kˆ are perpendicular to each other.. [C.B.S.E. 2000] Find a unit vector perpendicular to both a 3ˆi ˆj 2kˆ and b 2ˆi 3ˆj kˆ . [C.B.S.E. 2000] Define a × b and prove that a b a b a . b tan , where is the angle between
the vectors a and b . [C.B.S.E. 2000] If three vectors a , b and c are such that a b c 0 , prove that a b b c c a [C.B.S.E. 2001] If a 4ˆi 3ˆj kˆ and b i 2k find 2b a . [C.B.S.E. 2001]
Q.6
Find a vector whose magnitude is 3 units and which is perpendicular to the following two vectors a and b : a 3ˆi ˆj 4kˆ and b 6ˆi 5ˆj 2kˆ [C.B.S.E. 2001]
Q.7
In OAB, OA 3ˆi 2ˆj k and OB ˆi 3ˆj k . Find the area of the triangle.[C.B.S.E. 2002]
Q.8
2 2 Prove that a b | a |2 .| b |2 a . b .
Q.9
For any two vectors a and b , show that 1 | a |2
Q.10
[C.B.S.E. 2002] If a , b and c are position vectors of points A, B and c, then prove that [C.B.S.E. 2002] a b b c c a is a vector perpendicular to the plane of ABC.
[C.B.S.E. 2002]
1 | b | 1 a.b
2
2
| a b a b |2
Q.11
Find the value of so that the two vectors 2ˆi 3ˆj kˆ and. 4ˆi 6ˆj kˆ are
Q.12
(i) Parallel (ii) Perpendicular to each other [C.B.S.E. 2003] Find the unit vector perpendicular to the plane ABC where the position vectors
Q.13 Q.14
Q.15
Q.16
Q.17
of A, B and C are 2ˆi ˆj kˆ, ˆi ˆj 2k and 2ˆi 3kˆ respectively.. [C.B.S.E. 2004] If a 5ˆi ˆj 3kˆ and b ˆi 3ˆj 5kˆ , then show that vectors a b and a b are orthogonal. [C.B.S.E. 2004] Show that the points whose position vectors are a 4ˆi 3ˆj kˆ. bˆ 2ˆi 4ˆj 5kˆ
and c ˆi ˆj form a right angled triangle. [C.B.S.E. 2005] Let a ˆi ˆj, b 3 j kˆ . Find a vector d which is perpendicular to both a and b and c . d =1. [C.B.S.E. 2005] Express the vector a 5ˆi 2ˆj 5kˆ as sum of two vectors such that one is parallel to the vector b 3 ˆi kˆ and the other is perpendicular to b . [C.B.S.E. 2005] If a , b and c are mutually perpendicular vectors of equal magnitude, show that they are equally inclined to the vector a b c . [C.B.S.E. 2006] If a ˆi 2ˆj 3kˆ and b 3ˆi ˆj 2kˆ , show that a b and a b are perpendicular to
Q.18
[C.B.S.E. 2006] Find the angle between the vectors a b and a b where a 2ˆi ˆj 3kˆ and b 3ˆi ˆj 2kˆ [C.B.S.E. 2006] ˆ ˆ ˆ If a i j k and b ˆj kˆ find a vector c such that a c b and a . c 3 . [C.B.S.E. 2007] each other.
Q.19 Q.20
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VECTORS
136
Find the projection of b c on a where a 2ˆi 2ˆj kˆ , b ˆi 2ˆj 2kˆ and c 2ˆi ˆj 4kˆ [C.B.S.E. 2007] Three vectors a , b and c satisfy the condition a b c 0 . Find the value of [C.B.S.E. 2008] a.b b.c c.a if a 1, b 4 and c 2 .
Q.21 Q.22
Find a vector of magnitude 5 units. perpendicular to each of the vectors a b and a b where a ˆi ˆj kˆ and b ˆi 2ˆj 3kˆ [C.B.S.E. 2008]
Q.23
Q.24
If ˆi ˆj kˆ , 2ˆi 5ˆj . 3ˆi 2ˆj 3kˆ and ˆi 6ˆj kˆ are the position vectors of the points A, B, C
Q.25
and D, find the angle between and CD . Deduce that and CD are collinear.. AB AB [C.B.S.E. 2008] If a b c 0 and a 3, b 5 , and c 7, show that angle between a and b is 60º.
[C.B.S.E. 2008] Q.26
The scalar product of the vector ˆi ˆj kˆ with a unit vector along the sum of vectors
2ˆi 4ˆj 5kˆ and is equal to one. Find the value of If P is a unit vector and x p x p 80 , then find x . Find the projection of a on b if a . b = 8 and b 2ˆi 6ˆj 3kˆ If a ˆi ˆj kˆ , b 4ˆi 2ˆj 3kˆ and c ˆi 2ˆj kˆ , find a vector of which is parallel to the vector 2a b 3c . Let a ˆi 4ˆj 2kˆ, b 3ˆi 2ˆj 7kˆ and c 2ˆi ˆj 4kˆ . Find a vector perpendicular to both a and b and c . d 18. ˆi 2ˆj 3kˆ
Q.27 Q.28 Q.29
Q.30
[C.B.S.E. 2009]
.
[C.B.S.E. 2009] [C.B.S.E. 2009] magnitude 6 units [C.B.S.E. 2010]
d which is [C.B.S.E. 2010]
Q.31
Using vectors, find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5). [C.B.S.E. 2011]
Q.32
Let a i 4 j 2 k , b 3 i 2 j 7 k and c 2 i j 4 k . Find a vector p which is
^
^
^
^
^
^
^
^
^
perpendicular to both a and b and p . c = 18. ^
Q.32
^
^
^
^
[C.B.S.E. 2012]
^
If a i j 7 k and b 5 i j k , then find the value of , so that a b and a b are perpendicular vectors[C.B.S.E. 2013]
ANSWER KEY EXERCISE – 2 (BOARD PROBLEMS) 1. –1
2.
1 5 3
(5 ˆi ˆj 7kˆ )
6. 2ˆi 2ˆj kˆ
7.
3 2
10 sq units
10. 60 27. 6 14
11. (i) –2
(ii) 26
1
12.
14
( 3 ˆi 2ˆj kˆ
15.
1 ( ˆi ˆj 3kˆ ) 4
16. 6 ˆi 2kˆ ; ˆi 2ˆj 3kˆ
19.
2
20.
5ˆ 2ˆ 2 ˆ i j k 3 3 3
21.
2 22. – 31.
5 10 5 21 8 ˆ – 23. – ˆ + ˆ 24. 180º 26. 1 27. 9 28. 6 i 6 j 6 k 2 7
1 61 2
32.
^
^
29. 2ˆi 4ˆj 4kˆ 30. 64ˆi 2ˆj 28kˆ
^
p 64 i 2 j 28 k
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