If $a, b$ and $c$ are position vectors of the vertices of $\triangle ABC$,then $\frac{(a-c) \times (b-a)}{(b-a) \cdot (c-a)} = $

If $2 \hat{i}+3 \hat{j}-4 \hat{k}$ and $-\hat{i}+2 \hat{j}+\hat{k}$ are the two diagonals of a parallelogram, then the area of the parallelogram in square units is

Find the magnitude of the torque of a couple formed by a force $\vec{F} = 3\hat{i} + 2\hat{j} - \hat{k}$ acting at the point $\hat{i} - \hat{j} + \hat{k}$ and the force $-\vec{F}$ acting at the point $2\hat{i} - 3\hat{j} - \hat{k}$.

Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$ be two vectors such that $|\vec{a}|=1$,$\vec{a} \cdot \vec{b}=2$,and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a} \times \vec{b})-3 \vec{b}$,then the angle between $\vec{b}$ and $\vec{c}$ is equal to:

If the position vectors of the vertices of $\triangle ABC$ are $3\hat{i}+4\hat{j}-\hat{k}$, $\hat{i}+3\hat{j}+\hat{k}$, and $5(\hat{i}+\hat{j}+\hat{k})$ respectively, then the magnitude of the altitude from $A$ onto the side $BC$ is

Let $\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}$,$\bar{b}=\hat{i}+\hat{j}$ and $\bar{c}$ be a vector such that $|\bar{c}-\bar{a}|=4$,$|(\bar{a} \times \bar{b}) \times \bar{c}|=3$ and the angle between $\bar{c}$ and $\bar{a} \times \bar{b}$ is $\frac{\pi}{6}$,then $\bar{a} \cdot \bar{c}$ is equal to