Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors. Prove that $|\vec{a}\times\vec{b}|\le|\vec{a}||\vec{b}|$ . State the condition under which equality holds, i.e., $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|$

If vectors $\vec{a}=3\hat{i}+2\hat{j}+\lambda\hat{k}$ and $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$, represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of $\lambda$ is :

If $\vec{AB}=\hat{j}+\hat{k}$ and $\vec{AC}=3\hat{i}-\hat{j}+4\hat{k}$ represent the two vectors along the sides AB and AC of $\Delta ABC$, prove that the median $\vec{AD}=\frac{\vec{AB}+\vec{AC}}{2}$ where D is midpoint of BC. Hence, find the length of median AD.

During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $\vec{B}=2\hat{i}+8\hat{j}$, $\vec{W}=6\hat{i}+12\hat{j}$ and $\vec{F}=12\hat{i}+18\hat{j}$ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.

The value of $(\hat{i}\times\hat{j})\cdot \hat{j}+(\hat{j}\times\hat{i}) \hat{k}:$