Let $u, v, w$ be vectors such that $u + v + w = 0$. If $|u| = 3, |v| = 4,$ and $|w| = 5,$ find the value of $u \cdot v + v \cdot w + w \cdot u$.

Let $\vec{a}$ and $\vec{b}$ be two non-collinear vectors. For what values of $x$ and $y$ is the equation $2\vec{u} - \vec{v} = \vec{w}$ true,where $\vec{u} = x\vec{a} + 2y\vec{b}$,$\vec{v} = -2y\vec{a} + 3x\vec{b}$,and $\vec{w} = 4\vec{a} - 2\vec{b}$?

If the length of a vector is $21$ and its direction ratios are $2, -3, 6$,then its direction cosines are:

If $\vec{a} = -4\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 12\hat{i} - 6\hat{j} + 15\hat{k}$,then the vectors $\vec{a}$ and $\vec{b}$ are $.......$

In triangle $ABC$ (Fig.),which of the following is not true:

If $P$ and $Q$ are two points on the curve $y = 2^{x+2}$ in the rectangular Cartesian coordinate system such that $\overline{OP} \cdot \hat{i} = -1$ and $\overline{OQ} \cdot \hat{i} = 2$,then $\overline{OQ} - 4\overline{OP} = $