Question #fca5a

Jul 23, 2017

(3)
$\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 3 \left(\vec{c} \times \vec{a}\right)$

Explanation:

Given that $\vec{a} + 2 \vec{b} + 3 \vec{c} = \vec{O}$

Now, Taking cross product or vector product with $\vec{a}$ on both sides:-

$\left(\vec{a} + 2 \vec{b} + 3 \vec{c}\right) \times \vec{a} = \vec{O} \times \vec{a}$

Since cross product is distributive and cross product with $\vec{O}$ OR null vector is $\vec{O}$ itself,

$\implies \vec{a} \times \vec{a} + 2 \left(\vec{b} \times \vec{a}\right) + 3 \left(\vec{c} \times \vec{a}\right) = \vec{O}$

$\because$ the cross product of a vector with itself is null vector or $\vec{O}$ $\therefore$ $\vec{a} \times \vec{a} = \vec{O}$

$\implies 2 \left(\vec{b} \times \vec{a}\right) + 3 \left(\vec{c} \times \vec{a}\right) = \vec{O}$ ------------------- 1.

Similarly, taking cross product with $\vec{b} \mathmr{and} \vec{c}$ on both sides:-

$\left(\vec{a} \times \vec{b}\right) + 3 \left(\vec{c} \times \vec{b}\right) = \vec{O}$ ------------ 2.

$\left(\vec{a} \times \vec{c}\right) + 2 \left(\vec{b} \times \vec{c}\right) = \vec{O}$ ------------ 3.

$2 \left(\vec{b} \times \vec{a}\right) + 3 \left(\vec{c} \times \vec{a}\right) + \left(\vec{a} \times \vec{b}\right) + 3 \left(\vec{c} \times \vec{b}\right) + \left(\vec{a} \times \vec{c}\right) + 2 \left(\vec{b} \times \vec{c}\right) = \vec{O}$

Now for any two vectors $\vec{x}$ and $\vec{y}$, $\textcolor{red}{\vec{x} \times \vec{y} = - \vec{y} \times \vec{x}}$.

Also, sum of any vector with null vector is the vector itself.

$\implies \left(\vec{a} \times \vec{b}\right) - 2 \left(\vec{a} \times \vec{b}\right) + 2 \left(\vec{b} \times \vec{c}\right) - 3 \left(\vec{b} \times \vec{c}\right) + 3 \left(\vec{c} \times \vec{a}\right) - \left(\vec{c} \times \vec{a}\right) = \vec{O}$

$\implies - \left(\vec{a} \times \vec{b}\right) - \left(\vec{b} \times \vec{c}\right) - \left(\vec{c} \times \vec{a}\right) + 3 \left(\vec{c} \times \vec{a}\right) = \vec{O}$

$\implies - \left(\vec{a} \times \vec{b}\right) - \left(\vec{b} \times \vec{c}\right) - \left(\vec{c} \times \vec{a}\right) = - 3 \left(\vec{c} \times \vec{a}\right)$

Taking product on both sides with $- 1$,

$\implies \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 3 \left(\vec{c} \times \vec{a}\right)$