vector

378 Vectors
3. Choose y in order that the vectors
^ ^ ^ ^ ^
a 7 i yj kˆ
, b 3 i 2 j k,
^ ^ ^
c 5 i 3 j k are linearly dependent. Ans. y = 4
4. Prove that
[ a b, b c, c a] 2[ a b c]
5.18 COPLANARITY QUESTIONS
Example 6. Find the volume of tetrahedron having vertices
^ ^ ^
^ ^ ^
^ ^ ^ ^ ^
( j k), ( 4i 5j qk), ( 3i 9j
4k ) and 4( i j k)
.
Also find the value of q for which these four points are coplanar.
(Nagpur University, Summer 2004, 2003, 2002)
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
Solution. Let A = j k, B4 i 5 jqk, C 3 i 9 j4 k, D 4( i j k)
AB =
^ ^ ^ ^ ^ ^ ^
B A 4 i 5 j qk ( jkˆ
) 4i 6 j( q
1) k
AC =
^ ^ ^ ^ ^ ^ ^
C A (3 i 9ˆj 4 k) ( jk) 3 i 10 j5
k
AD = ^ ^ ^ ^ ^ ^ ^ ^
D A 4( i jk) ( jk) 4i 5 j
5k
Volume of the tetrahedron = 1 [ AB AC AD]
6
4 6 q 1
1
= 3 10 5 = 1 {4(50 25) 6(15 20) ( q 1)(15 40)}
6
6
4 5 5
= 1 {100 210 55 ( q 1)} = 1 ( 110 55 55 q)
6
6
= 1 ( 5555 q) 55 ( q1)
6 6
If four points A, B, C and D are coplanar, then ( AB AC AD ) = 0
i.e., Volume of the tetrahedron = 0
55
( q 1) = 0 q = 1 Ans.
6
Example 7. If four points whose position vectors are a, b, c,
d are coplanar, show that
[ a b c] [ a d b] [ a d c] [ d b c ] (Nagpur University, Summer 2005)
Solution. Let A, B, C, D be four points whose position vectors are a, b, c,
d .
AD = d a, BD d b and CD d
c
If AD, BD,
CD are coplanar, then
AD .( BD CD ) = 0
( d a) .[( d b) ( d c )] = 0
( d a) .[ d d d c b d b c ] = 0
( d a) .[ d c b d b c ] = 0
d .( d c) d .( b d) d .( b c) a .( d c) a .( b a) a .( b c ) = 0
0 0 [ dbc] [ ddc] [ dbd] [ abc ] = 0
[ abc ] [ abd] [ adc] [ dbc]
Proved.