Product of Vectors

A vector of magnitude $9$ and perpendicular to both the vectors $4\hat{\mathrm{i}}-\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ & $-2\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\hat{\mathrm{k}}$ is:

Let  $\overrightarrow{a}=\widehat{i}+4\widehat{j}+2\widehat{k}, \overrightarrow{b}=3\widehat{i}-2\widehat{j}+7\widehat{k}$ and  $\overrightarrow{c}=2\widehat{i}-\widehat{j}+4\widehat{k}$ . Which of the following is representing a vector $\overrightarrow{p}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and also whose scalar product with vector $\overrightarrow{c}$ would be $\overrightarrow{p}.\overrightarrow{c}=18.$

Two projectiles are fired from the same point with the same speed at angles of projection $60°\mathrm{and}30°$ respectively. The correct statement is

$\overrightarrow{a}=2\hat{i}+3\hat{j}+4\hat{k} \& \overrightarrow{b}=4\hat{i}+2\hat{j}+3\hat{k}$. Prove Cauchy-Schwarz inequality for vectors.

$\overrightarrow{OA}=4\hat{i}+3\hat{j}+5\hat{k}$, $\overrightarrow{OB}=2\hat{i}+3\hat{j}+2\hat{k}$ and $\overrightarrow{BA}=\hat{i}+3\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

$\overrightarrow{OA}=4\hat{i}+3\hat{j}+5\hat{k}$, $\overrightarrow{OB}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{BA}=3\hat{i}+\hat{j}+2\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

$\overrightarrow{OA}=3\hat{i}+2\hat{j}+5\hat{k}$, $\overrightarrow{OB}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{BA}=2\hat{i}+2\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

$\overrightarrow{OA}=3\hat{i}+2\hat{j}+5\hat{k}$, $\overrightarrow{OB}=4\hat{i}+5\hat{j}+2\hat{k}$ and $\overrightarrow{BA}=-\hat{i}-3\hat{j}+3\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

$\overrightarrow{OA}=2\hat{i}+3\hat{j}+4\hat{k}$, $\overrightarrow{OB}=4\hat{i}+5\hat{j}+2\hat{k}$ and $\overrightarrow{BA}=-2\hat{i}-2\hat{j}+2\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.