If $\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k}, \vec{b} = - \hat{i} + \hat{k}, \vec{c} = 2 \hat{j} - \hat{k}$ are three vectors, find the area of the parallelogram having diagonals $(\vec{a} + \vec{b})$ and $(\vec{b} + \vec{c})$ . [CBSE 2014]

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Solution

It is given that $\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k}, \vec{b} = - \hat{i} + \hat{k}, \vec{c} = 2 \hat{j} - \hat{k}$ .

∴ $\vec{a} + \vec{b} = (2 \hat{i} - 3 \hat{j} + \hat{k}) + (- \hat{i} + \hat{k}) = \hat{i} - 3 \hat{j} + 2 \hat{k}$

$\vec{b} + \vec{c} = (- \hat{i} + \hat{k}) + (2 \hat{j} - \hat{k}) = - \hat{i} + 2 \hat{j}$

We know that the area of parallelogram is $\frac{1}{2} |\vec{d_{1}} \times \vec{d_{2}}|$ , where $\vec{d_{1}}$ and $\vec{d_{2}}$ are the diagonal vectors.

Now,

$(\vec{a} + \vec{b}) \times (\vec{b} + \vec{c}) = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 3 & 2 \\ - 1 & 2 & 0 \end{matrix}| = - 4 \hat{i} - 2 \hat{j} - \hat{k}$

∴ Area of the parallelogram having diagonals $(\vec{a} + \vec{b})$ and $(\vec{b} + \vec{c})$

$= \frac{1}{2} |(\vec{a} + \vec{b}) \times (\vec{b} + \vec{c})| = \frac{1}{2} |- 4 \hat{i} - 2 \hat{j} - \hat{k}| = \frac{1}{2} \sqrt{{(- 4)}^{2} + {(- 2)}^{2} + {(- 1)}^{2}} = \frac{\sqrt{21}}{2} square units$

Thus, the required area of the parallelogram is $\frac{\sqrt{21}}{2}$ square units.

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Similar questions

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

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

¯ ¯ ¯ a + ¯ ¯ b

¯ ¯ b + ¯ ¯ c

[\vec{a} \vec{b} \vec{c}]

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

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

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

\vec{a} . (\vec{b} \times \vec{c})

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

\vec{c} = 3 \hat{i} + \hat{j} + 2 \hat{k}

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

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

\vec{a} = 11 \hat{i}, \vec{b} = 2 \hat{j}, \vec{c} = 13 \hat{k}

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

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