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  4. The three vectors hat i + hat j , hat j + hat k , hat k + hat i taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume :

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Q. The three vectors $\hat{ i }+\hat{ j }, \hat{ j }+\hat{ k }, \hat{ k }+\hat{ i }$ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume :

Vector Algebra

Solution:

$(\hat{i}+\hat{j}) \times(\hat{j}+\hat{k})=\hat{i}-\hat{j}+\hat{k} ;$ so the unit vector $\perp$ to the
plane of $\hat{ i }+\hat{ j }$ and $\hat{ j }+\hat{ k }$ is $\frac{1}{\sqrt{3}}(\hat{ i }-\hat{ j }+\hat{ k }) \cdot$ Similarly, the
other two unit vectors are $\frac{1}{\sqrt{3}}(\hat{ i }+\hat{ j }-\hat{ k })$ and
$\frac{1}{\sqrt{3}}(-\hat{ i }+\hat{ j }+\hat{ k })$
Hence, the required volume $=\frac{1}{3 \sqrt{3}}\left|\begin{array}{ccc}1 & -1 & 1 \\ 1 & 1 & -1 \\ -1 & 1 & 1\end{array}\right|=\frac{4}{3 \sqrt{3}}$