If $(x_1,x_2,x_3)\times (1,3,1)=(2,1,6)$ then find $ (x_1,x_2,x_3).$

Question

If $(x_1,x_2,x_3)\times (1,3,1)=(2,1,6)$ then find $ (x_1,x_2,x_3).$ Here $\times$ denotes the vector product.

The choices of answer are:

$A)(0,0,1)$

$B)(-1,2,7)$

$C)(-m,0,1-m)$ for all real $m$

$D)$There does not exist any expression $(x_1,x_2,x_4)$ in $\mathbb R^3.$

Answer 1

Note that by definition of cross product if

$$\vec w = \vec v \times \vec u \implies \vec w \perp \vec v,\vec u \iff \vec w\cdot\vec v=0,\quad\vec w\cdot\vec u=0$$

but in this case the dot product

$$(1,3,1)\cdot(2,1,6)=2+3+6=11\neq0$$

thus there does not exist any expression $(x_1,x_2,x_4)$ in $\mathbb{R^3}$ such that

$$(x_1,x_2,x_3)\times (1,3,1)=(2,1,6)$$

Answer 2

Hint: Given two vectors $v$ and $w$, then $v\times w$ is orthogonal to both $v$ and $w$.