Question

a b and c vector are non zero and non coplannar then find value of a vector ×​

Answer 1

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