If $\vec{f}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{g}=2 \hat{i}-\hat{j}+3 \hat{k}$,then the projection vector of $\vec{f}$ on $\vec{g}$ is

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\bar{c}=\hat{a}+2 \hat{b}$ and $\bar{d}=5 \hat{a}-4 \hat{b}$ are perpendicular to each other,then the angle between $\hat{a}$ and $\hat{b}$ is

If the angle between the vectors $a$ and $b$ is $\theta$ and $a \cdot b = \cos \theta$,then the true statement is:

Let $ABCD$ be a parallelogram where $\overrightarrow{AB} = \overrightarrow{a}$,$\overrightarrow{AD} = \overrightarrow{b}$,$|\overrightarrow{a}| = |\overrightarrow{b}| = 2$ and $|\overrightarrow{a} \times \overrightarrow{b}| + \overrightarrow{a} \cdot \overrightarrow{b} = \sqrt{2} |\overrightarrow{a}| |\overrightarrow{b}|$,where $\overrightarrow{a} \cdot \overrightarrow{b} > 0$. Then the area of this parallelogram is (in square units):

If $\vec{f}, \vec{g}, \vec{h}$ are mutually orthogonal vectors of equal magnitudes,then the angle between the vectors $\vec{f}+\vec{g}+\vec{h}$ and $\vec{h}$ is

Let $a$ and $b$ be linearly independent vectors such that $|a| = \sqrt{3}$, $|b| = 3$, and $|a - b| = 4$. If $a \times (2i + 2j - k) = (2i + 2j - k) \times b$ and $|(a + b) \cdot (3i + 4j + 2k)| = \sqrt{\lambda}$, then $\lambda = ...$