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Find the Area of a Parallelogram Whose Adjacent Sides Are Represented by the Vectors 2 ^ I − 3 ^ K and 4 ^ J + 2 ^ K . - Mathematics

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Find the area of a parallelogram whose adjacent sides are represented by the vectors\[2 \hat{i} - 3 \hat{k} \text { and } 4 \hat{j} + 2 \hat{k} .\]

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Solution

\[\text { Let } \vec{a} = 2 \hat{i} - 3 \hat{k} \text { and } \vec{b} = 4 \hat {j}+ 2 \hat{k}. \]

\[ \vec{a} \times \vec{b} = \begin{vmatrix}i & j & k \\ 2 & 0 & - 3 \\ 0 & 4 & 2\end{vmatrix}\]

\[ = 12 \hat{i} - 4 \hat{j} + 8 \hat{k} \]

\[\text { Thus }, \left| \vec{a} \times \vec{b} \right| = \sqrt{\left( 12 \right)^2 + \left( - 4 \right)^2 + \left( 8 \right)^2} = \sqrt{224} = 4\sqrt{14}\]

∴ Area of the parallelogram = \[\left| \vec{a} \times \vec{b} \right| = 4\sqrt{14}\] square units

Concept: Vector Joining Two Points
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