Given $p = 2a - 3b$,$q = a - 2b + c$,and $r = -3a + b + 2c$,where $a, b,$ and $c$ are non-zero,non-coplanar vectors,then the vector $-2a + 3b - c$ is equal to:

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3} \hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in the counterclockwise direction in the first quadrant. Then the area of the triangle having vertices $(\alpha, \beta), (0, \beta)$ and $(0,0)$ is equal to

If $P(2, 3, 1)$ and $Q(7, 15, 1)$,then $|\overrightarrow{PQ}| = \dots$

Find a vector of magnitude $5$ units,and parallel to the resultant of the vectors $\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+\hat{k}.$

If $ABCD$ is a parallelogram with $AC$ and $BD$ as diagonals,then which of the following is true regarding the vector relationship between the diagonals and sides?

Classify the following as scalar or vector quantities:
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