First slide

Scalar triple product of vectors

Question

$Let the volume of a parallelepiped whose coterminous edges are given by$ $\begin{array}{l} \vec{u} = \hat{i} + \hat{j} + λ \hat{k}, \vec{v} = \hat{i} + \hat{j} + 3 \hat{k}, and \vec{w} = 2 \hat{i} + \hat{j} + \hat{k}, be 1 cu.unit. If θ be the angle between the \\ edges \vec{u} and \vec{w}, then \cos θ can be: \end{array}$

Moderate

A

$\frac{7}{6 \sqrt{3}}$
B

$\frac{7}{6 \sqrt{6}}$
C

$\frac{5}{7}$
D

$\frac{5}{3 \sqrt{3}}$

Solution

$\begin{array}{l} Volume of parallelepiped formed by coterminous vectors \bar{a}, \bar{b}, \vec{c} is [\bar{a} \bar{b} \vec{c}] \\ ∴ \pm 1 = |\begin{array}{l} 1 & 1 & λ \\ 1 & 1 & 3 \\ 2 & 1 & 1 \end{array}| \Rightarrow = - λ + 3 = \pm 1 \Rightarrow λ = 2 or λ = 4 \\ For λ = 2, \cos θ = \frac{2 + 1 + 2}{\sqrt{6} \sqrt{6}} = \frac{5}{6} \\ For λ = 4, \cos θ = \frac{2 + 1 + 4}{\sqrt{6} \sqrt{18}} = \frac{7}{6 \sqrt{3}} \end{array}$

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