Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram...

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}+2 \vec{b}$ and $3 \vec{a}-\lambda \vec{b}$ are...

Let ${{\text{L}}_{1}}:\frac{{x-1}}{2}=\frac{{y-2}}{3}=\frac{{z-3}}{4}$ and ${{\text{L}}_{2}}:\frac{{x-2}}{3}=\frac{{y-4}}{4}=\frac{{z-5}}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between ${{\text{L}}_{1}}$ and ${{\text{L}}_{2}}$...