Question

Assertion :If $a,b,c$ are three non-coplanar, non-zero vectors, then $(a.a)b\times c+(a.b)c\times a+(a.c)a\times b=\left[bca\right]a$ Reason: If the vectors $a,b,c$ are non-coplanar, then so are $b\times c,c\times a,a\times b$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

As $a,b,c$ are non-coplanar, $b\times c,c\times a,a\times b$ are also non-coplanar.

So, any vector can be expressed as a linear combination of these vectors.

Let $a=\lambda b\times c+\mu c\times a+va\times b$

$\therefore a.a=\lambda \left[bca\right],a.b=\mu \left[cab\right],a.c=v\left[abc\right]$

$\therefore a=\frac{(a.a)b\times c}{\left[bca\right]}+\frac{(a.b)c\times a}{\left[cab\right]}+\frac{(a.c)a\times b}{\left[abc\right]}$

$\therefore (a.a)b\times c+(a.b)c\times a+(a.c)a\times b=\left[bca\right]a.$