Solveeit Logo
Login

Question

Question: Let \(a,b,c\) be three vectors from \(a \times (b \times c) = (a \times b) \times c\), if...

Let a,b,ca,b,c be three vectors from a×(b×c)=(a×b)×ca \times (b \times c) = (a \times b) \times c, if

A

b×(a×c)=0b \times (a \times c) = 0

B

a(b×c)=0a(b \times c) = 0

C

c×a=a×bc \times a = a \times b

D

c×b=b×ac \times b = b \times a

Answer

b×(a×c)=0b \times (a \times c) = 0

Explanation

Solution

We have a×(b×c)=(a×b)×c\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}

⇒(a.c)b−(a.b)c=(a.c)b−(b.c)a\Rightarrow (\mathbf{a}.\mathbf{c})\mathbf{b} - (\mathbf{a}.\mathbf{b})\mathbf{c} = (\mathbf{a}.\mathbf{c})\mathbf{b} - (\mathbf{b}.\mathbf{c})\mathbf{a}

⇒−(a.b)c=−(b.c)a⇒(b.c)a−(b.a)c=0\Rightarrow - (\mathbf{a}.\mathbf{b})\mathbf{c} = - (\mathbf{b}.\mathbf{c})\mathbf{a} \Rightarrow (\mathbf{b}.\mathbf{c})\mathbf{a} - (\mathbf{b}.\mathbf{a})\mathbf{c} = 0

⇒b×(a×c)=0.\mathbf{\Rightarrow}\mathbf{b}\mathbf{\times}\mathbf{(a}\mathbf{\times}\mathbf{c) = 0.}