Find the sine of the angle between the vectors $\vec{a}=3 \hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}-2 \hat{j}+4 \hat{k}$.

Let $\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ with $a_{i} > 0$ for $i = 1, 2, 3$ be a vector that makes equal angles with the coordinate axes $OX$,$OY$,and $OZ$. Also,let the projection of $\vec{a}$ on the vector $3 \hat{i} + 4 \hat{j}$ be $7$. Let $\vec{b}$ be a vector obtained by rotating $\vec{a}$ by $90^{\circ}$. If $\vec{a}$,$\vec{b}$,and the $x$-axis are coplanar,then the projection of vector $\vec{b}$ on $3 \hat{i} + 4 \hat{j}$ is equal to

Let $a, b, c$ be three vectors such that $a \neq 0$,$a \times b = 2a \times c$,$|a| = |c| = 1$,$|b| = 4$,and $|b \times c| = \sqrt{15}$. If $b - 2c = \lambda a$,then $\lambda$ equals:

Let $v_1, v_2, v_3, v_4$ be unit vectors in the $XY$-plane,one each in the interior of the four quadrants. Which of the following statements is necessarily true?

If the angle between two vectors $\vec{u} = i + k$ and $\vec{v} = i - j + ak$ is $\pi / 3,$ then the value of $a$ is:

Let $\overrightarrow{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\overrightarrow{b} = 7\hat{i} + \hat{j} - 6\hat{k}$. If $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{r} \times \overrightarrow{b}$ and $\overrightarrow{r} \cdot (\hat{i} + 2\hat{j} + \hat{k}) = -3$,then $\overrightarrow{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k})$ is equal to: