Let $a = i - k$, $b = xi + j + (1 - x)k$, and $c = yi + xj + (1 + x - y)k$. Then $[a\,b\,c]$ depends on

The value of $[\vec{a}-\vec{b} \quad \vec{b}-\vec{c} \quad \vec{c}-\vec{a}]$ is equal to

If the vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$,$\vec{b}=\hat{i}+2 \hat{j}-3 \hat{k}$,and $\vec{c}=3 \hat{i}+p \hat{j}+5 \hat{k}$ are coplanar,then $p=$

Let $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$ be vectors in three-dimensional space,where $\overrightarrow{u}$ and $\overrightarrow{v}$ are unit vectors which are not perpendicular to each other and $\overrightarrow{u} \cdot \overrightarrow{w}=1, \overrightarrow{v} \cdot \overrightarrow{w}=1, \overrightarrow{w} \cdot \overrightarrow{w}=4$. If the volume of the parallelepiped,whose adjacent sides are represented by the vectors $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$,is $\sqrt{2}$,then the value of $|3\vec{u}+5\vec{v}|$ is.

The vector $c \cdot (b+c) \times (a+b+c)$ is equal to

Which of the following is not true?