Question
What is a vector of unit length orthogonal to both the vectors $$\hat i + \hat j + \hat k$$ and $$2\hat i + 3\hat j - \hat k\,?$$
A.
$$\frac{{ - 4\hat i + 3\hat j - \hat k}}{{\sqrt {26} }}$$
B.
$$\frac{{ - 4\hat i + 3\hat j + \hat k}}{{\sqrt {26} }}$$
C.
$$\frac{{ - 3\hat i + 2\hat j - \hat k}}{{\sqrt {14} }}$$
D.
$$\frac{{ - 3\hat i + 2\hat j + \hat k}}{{\sqrt {14} }}$$
Answer :
$$\frac{{ - 4\hat i + 3\hat j + \hat k}}{{\sqrt {26} }}$$
Solution :
\[\begin{array}{l}
\overrightarrow A = \hat i + \hat j + \hat k\\
\overrightarrow B = 2\hat i + 3\hat j - \hat k\\
\overrightarrow A \times \overrightarrow B = \left| \begin{array}{l}
\hat i\,\,\,\,\hat j\,\,\,\,\,\,\,\,\hat k\\
1\,\,\,\,\,1\,\,\,\,\,\,\,\,1\\
2\,\,\,\,3\,\, - 1
\end{array} \right|\\
= \hat i\left( { - 1 - 3} \right) - \hat j\left( { - 1 - 2} \right) + \hat k\left( {3 - 2} \right)\\
= - 4\hat i + 3\hat j + \hat k
\end{array}\]
Vector of unit length orthogonal to both the vectors $$\overrightarrow A $$ and $$\overrightarrow B $$
$$\eqalign{
& = \frac{{\overrightarrow A \times \overrightarrow B }}{{\left| {\overrightarrow A \times \overrightarrow B } \right|}} \cr
& = \frac{{ - 4\hat i + 3\hat j + \hat k}}{{\sqrt {16 + 9 + 1} }} \cr
& = \frac{{ - 4\hat i + 3\hat j + \hat k}}{{\sqrt {26} }} \cr} $$