List of top Higher Mathematics Questions on Vector Algebra

Find the area of a parallelogram whose adjacent sides are given by the vectors \( \mathbf{a} = \hat{i} - \hat{j} + 3 \hat{k} \) and \( \mathbf{b} = 2 \hat{i} - 7 \hat{j} + \hat{k} \).
Find the projection of vector \( \mathbf{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} \) on the vector \( \mathbf{b} = 2 \hat{i} + \hat{k} \).
Show that for any two vectors \( \mathbf{a} \) and \( \mathbf{b} \), always \( | \mathbf{a} \cdot \mathbf{b} | \leq \|\mathbf{a}\| \|\mathbf{b}\| \).
Find the value of \( x \) for which \( x ( \hat{i} + \hat{j} + \hat{k} ) \) is a unit vector.
If direction ratios of a line are -18, 12, -4, then find its direction cosines.
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 2\hat{k} \).
The direction cosines of \( x \), \( y \), and \( z \)-axis are \(\underline{\hspace{2cm}}\) respectively.