List of top Mathematics Questions on Scalar and Vector Product

If for two unit vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} + 2\vec{b}| = |2\vec{a} - \vec{b}| \), then find the angle between \( \vec{a} \) and \( \vec{b} \).

For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true ?

If \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198 \) and \( |\vec{a}| = 10 |\vec{b}| \), then :

Assertion (A): $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.
Reason (R): $|\vec{a} \times \vec{b}| = (\vec{a} \cdot \vec{b})\tan\theta$, where $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ ($\theta \neq \frac{\pi}{2}$).

If the direction vectors of two straight lines are given by \( \vec{m} = \hat{i} + \hat{j} + 0\hat{k} \) and \( \vec{n} = \hat{i} + 0\hat{j} + \hat{k} \), calculate the value of \( \cos\phi \) representing the acute angle between them.

The total area of a parallelogram constructed with adjacent vector sides given by \( \vec{a} = \hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - 7\hat{j} + \hat{k} \) is equal to:

A unit vector perpendicular to the vectors \[ \hat i-\hat j \] and \[ \hat i+\hat j \] is: