Word Count : 472

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We need to prove that the area of a parallelogram whose diagonals are given by vectors \(\vec{a}\) and \(\vec{b}\) is \[ \frac{|\vec{a} \times \vec{b}|}{2} \] ### Step 1: Proof Let the parallelogram be defined by two vectors \(\vec{p}\) and \(\vec{q}\). The diagonals of the parallelogram are given by: \[ \vec{a} = \vec{p} + \vec{q}, \quad \vec{b} = \vec{p} - \vec{q} \] From vector addition, we express the sides as: \[ \vec{p} = \frac{\vec{a} + \vec{b}}{2}, \quad \vec{q} = ___ __________ _______ ___ ___ ____ ______ ______ ___ ________ _________.
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