Question: 1. If a, b, c are three non-zero non coplanar vectors. Prove that 2a 3b 4c ,
3a 4b 5c , 2a 4b 6c are coplanar vectors.
Question: 2. If a, b, c are three non-zero non coplanar vectors. Prove that 3a 2b 4c ,
2a 2b 3c , 2a 2b 4c are non coplanar vectors.
Question: 3. Let a
2i 2 j 2k , b
2 j, c
i
k . Then if
j
2,
3 , find
which makes
a , b and c coplanar.
Question: 4. The volume of the parallelepiped whose edges are 4i 5 j 6k , 2i 3 j 4k and
2i
j
k is 14 cubic units. Find the value of
.
Question: 5. Show that the four points A(0,5,-2), B(4,3,-4), C(-2,9,-4) and D(-2,3,2) are
coplanar.
Question: 6. Show that the four points having the position vectors 5i 6 j 2k , i ,
4i 10 j 5k and
3i 5 j 5k are coplanar.
Question: 7. Show that the four points A(-2,3,-4), B(2,1,-6), C(-4,7,-6) and D(-4,1,0) are
coplanar.
Question: 8. Let V
3i 2 j 2k and W
2i 4k .If U is a unit vector, then find the maximum
value of the scalar triple product U , V , W
Question: 9. If the three vectors a
then find the value of
4i 2 j 2k , b
2i 2 j
k are coplanar
.
Question: 10. If the three vectors a
then find the value of
2i 4 j 2k , c
4i
j 2k , b
i 4 j 3k , c
8i
j 3k are coplanar
.
Question: 11. If a, b, c are three vectors the prove that [2a 2b, 2b 2c, 2c 2a] 16[a, b, c]
Question: 12. Find b.[(c a) (a b c)]
Question: 13. Find the value of
2i 2 j 2k , i 4 j
so that the four points with position vectors i 2k , 3i ,
k are coplanar.
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Question: 14. Find b.(c a) (b 2c 3a)
Question: 15. Find the volume of the parellelopiped whose edges are represented by the
vectors: a 3i 2 j 2k , b
2i 3k , c 3i
j
Question: 16. If a, b, c are three vectors then find [3a b,3b c,3c a]
Question: 17. If a, b, c are three non-coplanar vectors, then prove that
(a b c)[(a b) (b c)]
3[a, b, c]
Question: 18. Determine [a, b, c] if
(i) a i 2 j k , b
2k , c
(ii) a
j k , c 3i 2 j
2i
j, b i
2i
j 2k
Question: 19. Prove that four points A(0,1,-3), B(-1,-4,1), C(1,2,-4), D(-1,-8,4) are
coplanar.
Question:20. Prove that (v w).[( w u ) (u v)] =0
Answers:
A3. 2
A4. 7
A8.
336
A9. 4
A10. 6
A12. 0
A13. 4
A14. [b, c, a]
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A15. 13 cubic units
A16. 26 [a, b, c]
A18. (i) 6
(ii)
1
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