Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is:

If \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\), then \(\vec{a}\) and \(\vec{b}\):

Let \(\theta\) be the angle between two unit vectors \(\hat{a}\) and \(\hat{b}\) such that \(\sin\theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to:

Let \(|\vec{a}|=5\) and \(-2\le\lambda\le1\). Then, the range of \(|\lambda\vec{a}|\) is:

(i) Complete the given figure to explain their entire movement plan along the respective vectors. (ii) Find vectors $\vec{AC}$ and $\vec{BC}$. (iii) (a) If $\vec{a} \cdot \vec{b} = 1$, distance of O to A is 1 km and that from O to B is 2 km, then find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Also, find $| \vec{a} \times \vec{b}|$. OR (iii) (b) If $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, then find a unit vector perpendicular to $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$.