Question:
Vectors \( \vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{k} \) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.
Vectors \( \vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{k} \) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.
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One diagonal is the sum, representing the vector from origin to opposite vertex.
The other diagonal is the difference, representing the vector connecting the endpoints of side vectors.
The other diagonal is the difference, representing the vector connecting the endpoints of side vectors.
Updated On: Sep 10, 2026
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Solution and Explanation
Concept:
• Parallelogram Law: If \( \vec{a} \) and \( \vec{b} \) are adjacent sides, then the diagonals are \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \).
Step 1: Find the first diagonal vector \( \vec{d_1} \)
\[ \vec{d_1} = \vec{a} + \vec{b} \] \[ = (3\hat{i} - 2\hat{j} + 2\hat{k}) + (\hat{i} + 0\hat{j} + 2\hat{k}) \] \[ = 4\hat{i} - 2\hat{j} + 4\hat{k} \]
Step 2: Find the second diagonal vector \( \vec{d_2} \)
\[ \vec{d_2} = \vec{a} - \vec{b} \] \[ = (3\hat{i} - 2\hat{j} + 2\hat{k}) - (\hat{i} + 0\hat{j} + 2\hat{k}) \] \[ = 2\hat{i} - 2\hat{j} + 0\hat{k} = 2\hat{i} - 2\hat{j} \]
Step 3: Calculate the lengths of the diagonals
Length of \( \vec{d_1} \): \[ | \vec{d_1} | = \sqrt{4^2 + (-2)^2 + 4^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \] Length of \( \vec{d_2} \): \[ | \vec{d_2} | = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
• Parallelogram Law: If \( \vec{a} \) and \( \vec{b} \) are adjacent sides, then the diagonals are \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \).
Step 1: Find the first diagonal vector \( \vec{d_1} \)
\[ \vec{d_1} = \vec{a} + \vec{b} \] \[ = (3\hat{i} - 2\hat{j} + 2\hat{k}) + (\hat{i} + 0\hat{j} + 2\hat{k}) \] \[ = 4\hat{i} - 2\hat{j} + 4\hat{k} \]
Step 2: Find the second diagonal vector \( \vec{d_2} \)
\[ \vec{d_2} = \vec{a} - \vec{b} \] \[ = (3\hat{i} - 2\hat{j} + 2\hat{k}) - (\hat{i} + 0\hat{j} + 2\hat{k}) \] \[ = 2\hat{i} - 2\hat{j} + 0\hat{k} = 2\hat{i} - 2\hat{j} \]
Step 3: Calculate the lengths of the diagonals
Length of \( \vec{d_1} \): \[ | \vec{d_1} | = \sqrt{4^2 + (-2)^2 + 4^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \] Length of \( \vec{d_2} \): \[ | \vec{d_2} | = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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