List of top Mathematics Questions on Product of Two Vectors asked in CBSE Class XII

For two vectors \(\vec{a}\) and \(\vec{b}\):  

Assertion (A): \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Reason (R): \[ |\vec{a}\times\vec{b}| = (\vec{a}\cdot\vec{b})\tan\theta, \quad \theta\neq\frac{\pi}{2}. \]

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}, \] then the values of \(p\) and \(q\) are: 

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 \(\vec{b}\).
If \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198 \) and \( |\vec{a}| = 10|\vec{b}| \), then :
If \( |\vec{a}| = 8 \), \( |\vec{b}| = 3 \) and \( |\vec{a} \times \vec{b}| = 12 \), then the value of \( |\vec{a} \cdot \vec{b}| \) is