If $a = i + 2j + 2k$ and $b = 3i + 6j + 2k,$ then a vector in the direction of $a$ and having magnitude as $|b|$ is

Let $u$ and $v$ be two vectors in a plane. Then any vector $w$ in the plane can be written as $w = au + bv$ for some scalars $a$ and $b$ if and only if

For any two vectors $\vec{a}$ and $\vec{b}$,which of the following statements is true?

If $\overline{AB} \parallel \overline{CD}$ and the position vectors of the four points $A, B, C,$ and $D$ are $2i + j, i - 3j, 3i + 2j,$ and $i + \lambda j$ respectively,then the value of $\lambda$ is .....

Find the unit vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$.

$A$ vector $\vec{a}$ has components $2p$ and $1$ with respect to a rectangular Cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If $\vec{a}$ has components $p+1$ and $1$ with respect to the new system,then: