a b and c vector are non zero and non coplannar then find value of a vector ×
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Step-by-step explanation:
Let a =l(b×c)+m(c×a)+n(a×b) ...(1)
We have to find the values of l,m and n.
Multiply both sides of (1) scalarly by a.
a.a=la.(b×c)+ma.(c×a)+na.(a×b)
⇒a.a=l[abc]+m[aca]+n[aab]=l[abc].
∴ Scalar triple product is zero when two vectors are equal
∴l=
[abc]
a.a
Similarly, multiplying both sides of (1) salarly by b and c, we get m=
(abc)
a.b
,n=
[abc]
a.c
∴a=
[abc]
a.a
(b×c)+
[abc]
a.b
(c×a)+
[abc]
a.c
(a×b)
Similarly we can write the values b and c as b=
[abc]
b.a
(b×c)+
[abc]
b.b
(c×a)+
[abc]
b.c
(a×b)
and c=
[abc]
c.a
(b×c)+
[abc]
c.b
(c×a)+
[abc]
c.c
(a×b).
∴a=
[bca]
(a.a)b×c
+
[cab]
(a.b)c×a
+
[abc]
(a.c)a×b
∴(a.a)b×c+(a.b)c×a+(a.c)a×b=[bca]a
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