The orthogonal projection vector of $\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}$ on $\vec{b} = \hat{i} - 2\hat{j} + \hat{k}$ is

$A$ vector coplanar with vectors $i + j$ and $j + k$ and parallel to the vector $2i - 2j - 4k$ is:

If vector $a$ has magnitude $5$ and points north-east,and vector $b$ has magnitude $5$ and points north-west,then $|a - b| = $

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}$,$\vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$,$\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$,respectively. Then which of the following statements is true?

For non-zero vectors $a$ and $b$, if $|a+b| < |a-b|$, then $a$ and $b$ are

If $\overline{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}$,$\overline{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\overline{c}=3 \hat{i}+\hat{j}$ such that $\overline{b}+\lambda \overline{a}$ is perpendicular to $\overline{c}$,then $\lambda$ is