Question

The unit vector perpendicular to each of the vectors $2i-j+k$ and $3i+4j-k$ is

• A. $-3i+5j+11k$
• B. $(-3i+5j+11k)/155$
• C. $(-3i+5j+11k)/\sqrt{\left(155\right)}$
• D. None of these

Answer 1

The correct option is C $(-3i+5j+11k)/\sqrt{\left(155\right)}$

We know that $a\times b$ is vector perpendicular to the plane of $a$ and $b$
Therefore unit vector perpendicular to the plane of $a$ and $b$ $=\frac{a\times b}{|a\times b|}$
Now $a\times b=(2i-j+k)\times (3i+4j-k)$
$=\left|\begin{array}{ccc}i& j& k\\ 2& -1& 1\\ 3& 4& -1\end{array}\right|=i(1-4)+j(3+2)+k(8+3)=-3i+5j+11k$
$\Rightarrow$ unit vector parallel to $a\times b$
$=\frac{a\times b}{|a\times b|}=\frac{-3i+5j+11k}{\sqrt{(3^2+5^2+{11}^2)}}$
$=\frac{-3i+5j+11k}{\sqrt{155}}$