List of top Mathematics Questions on Vectors asked in KEAM

Let \(\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{j} + \alpha\hat{k}\). If \(|\vec{a} + \vec{b}| = \sqrt{29}\). Then the possible values of \(\alpha\) are

If $ |\vec{a}| = 5 $, $ |\vec{b}| = 8 $, $ |\vec{a} - \vec{b}| = 7 $, find the angle between  $ \vec{a} $ and $ \vec{b} $.

Let \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) be vectors such that \( \mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0} \). If \( |\mathbf{u}| = 3 \), \( |\mathbf{v}| = 4 \), and \( |\mathbf{w}| = 5 \), then \( \mathbf{u} \cdot \mathbf{v} + \mathbf{w} \cdot \mathbf{u} \) is:
Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors with magnitudes 4, 4, and 2, respectively. If \( \vec{a} \) is perpendicular to \( (\vec{b} + \vec{c}) \), \( \vec{b} \) is perpendicular to \( (\vec{c} + \vec{a}) \), and \( \vec{c} \) is perpendicular to \( (\vec{a} + \vec{b}) \), then the value of \( |\vec{a} + \vec{b} + \vec{c}| \) is:
If two vectors \( \vec{a} = \cos \alpha \hat{i} + \sin \alpha \hat{j} + \sin \frac{\alpha}{2} \hat{k} \) and \( \vec{b} = \sin \alpha \hat{i} - \cos \alpha \hat{j} + \cos \frac{\alpha}{2} \hat{k} \) are perpendicular, then the values of \( \alpha \) are: