What is the area of the parallelogram whose sides are represented by the vectors \hat{i}+2\hat{j}+3\hat{k} and 2\hat{i}+\hat{j}+2\hat{k}?

Explanation

The area of the parallelogram is given by the magnitude of the cross product of its adjacent sides. Let \vec{a} = \hat{i}+2\hat{j}+3\hat{k} and \vec{b} = 2\hat{i}+\hat{j}+2\hat{k}. Then \vec{a} \times \vec{b} = \hat{i}(4-3) - \hat{j}(2-6) + \hat{k}(1-4) = \hat{i} + 4\hat{j} - 3\hat{k}. Its magnitude is \sqrt{1^2 + 4^2 + (-3)^2} = \sqrt{1 + 16 + 9} = \sqrt{26} square units.

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