Given $\vec{a} = \frac{1}{7} (2 \hat{i} + 3 \hat{j} + 6 \hat{k}), \vec{b} = \frac{1}{7} (3 \hat{i} - 6 \hat{j} + 2 \hat{k}), \vec{c} = \frac{1}{7} (6 \hat{i} + 2 \hat{j} - 3 \hat{k}), \hat{i}, \hat{j}, \hat{k}$ being a right handed orthogonal system of unit vectors in space, show that $\vec{a}, \vec{b}, \vec{c}$ is also another system.

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Solution

$Given : \vec{a} = \frac{1}{7} (2 \hat{i} + 3 \hat{j} + 6 \hat{k}) \vec{b} = \frac{1}{7} (3 \hat{i} - 6 \hat{j} + 2 \hat{k}) \vec{c} = \frac{1}{7} (6 \hat{i} + 2 \hat{j} - 3 \hat{k}) \vec{a} \times \vec{b} = (\frac{1}{7}) (\frac{1}{7}) |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 6 \\ 3 & - 6 & 2 \end{matrix}| = \frac{1}{49} (42 \hat{i} + 14 \hat{j} - 21 \hat{k}) = \frac{1}{49} [7 (6 \hat{i} + 2 \hat{j} - 3 \hat{k})] = \frac{1}{7} (6 \hat{i} + 2 \hat{j} - 3 \hat{k}) = \vec{c} \vec{b} \times \vec{c} = (\frac{1}{7}) (\frac{1}{7}) |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & - 6 & 2 \\ 6 & 2 & - 3 \end{matrix}| = \frac{1}{49} (14 \hat{i} + 21 \hat{j} + 42 \hat{k}) = \frac{1}{49} [7 (2 \hat{i} + 3 \hat{j} + 6 \hat{k})] = \frac{1}{7} (2 \hat{i} + 3 \hat{j} + 6 \hat{k}) = \vec{a} \vec{c} \times \vec{a} = (\frac{1}{7}) (\frac{1}{7}) |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 2 & - 3 \\ 2 & 3 & 6 \end{matrix}| = \frac{1}{49} (21 \hat{i} - 42 \hat{j} + 14 \hat{k}) = \frac{1}{49} [7 (3 \hat{i} - 6 \hat{j} + 2 \hat{k})] = \frac{1}{7} (3 \hat{i} - 6 \hat{j} + 2 \hat{k}) = \vec{b} |\vec{a}| = \frac{1}{7} \sqrt{4 + 9 + 36} = \frac{7}{7} = 1 |\vec{b}| = \frac{1}{7} \sqrt{9 + 36 + 4} = \frac{7}{7} = 1 |\vec{c}| = \frac{1}{7} \sqrt{36 + 4 + 9} = \frac{7}{7} = 1 Thus, \vec{a}, \vec{b} and \vec{c} form a right handed orthogonal system of unit vectors.$

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Similar questions

\vec{a} = \frac{1}{7} (2 \hat{i} + 3 \hat{j} + 6 \hat{k}), \vec{b} = \frac{1}{7} (3 \hat{i} - 6 \hat{j} + 2 \hat{k}), \vec{c} = \frac{1}{7} (6 \hat{i} + 2 \hat{j} - 3 \hat{k})

\vec{a} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{b} = 3 \hat{i} + 2 \hat{j} + 7 \hat{k}, \vec{c} = 5 \hat{i} + 6 \hat{j} + 5 \hat{k}

\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k}, \vec{b} = - \hat{i} + 4 \hat{j} + 3 \hat{k}, \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}

\hat{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, \hat{b} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}, \hat{c} = \hat{i} - 3 \hat{j} + 5 \hat{k}

¯ ¯ ¯ a = 2 ¯ i + 3 ¯ j + 6 ¯ ¯ ¯ k, ¯ ¯ ¯ b = 3 ¯ i - 6 ¯ j + 2 ¯ ¯ ¯ k, ¯ ¯ ¯ c = 6 ¯ i + 2 ¯ j - 3 ¯ ¯ ¯ k

¯ ¯ ¯ a \times ¯ ¯ ¯ b =

A unit vector in the plane of the vectors 2i + j + k, i - j + k and orthogonal to 5i + 2j + 6k is

\vec{a} = \hat{i} - \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - \hat{k}, \vec{c} = λ \hat{i} - \hat{j} + λ \hat{k}

\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - 3 \hat{k}, \vec{c} = λ \hat{i} + λ \hat{j} + 5 \hat{k}

\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}, \vec{b} = 3 \hat{i} + λ \hat{j} + \hat{k}, \vec{c} = \hat{i} + 2 \hat{j} + 2 \hat{k}

\vec{a} = \hat{i} + 3 \hat{j}, \vec{b} = 5 \hat{k}, \vec{c} = λ \hat{i} - \hat{j}

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