Vector Algebra

If $\vec a$ and $\vec b$ are vectors in space given by $\vec a=\dfrac{\hat i-2\hat j}{\sqrt5}$ and $\vec b=\dfrac{2\hat i+\hat j+3\hat k}{\sqrt{14}}$, then the value of $(2\vec a+\vec b)\cdot\left[(\vec a\times\vec b)\times(\vec a-2\vec b)\right]$ is

Two adjacent sides of a parallelogram ABCD are given by $\overrightarrow{AB}=2\hat i+10\hat j+11\hat k$ and $\overrightarrow{AD}=-\hat i+2\hat j+2\hat k$.

The side AD is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that AD becomes AD′. If AD′ makes a right angle with the side AB, then the cosine of the angle $\alpha$ is given by

Match the statements in Column-I with the values in Column-II.

If $\vec a,\vec b,\vec c$ and $\vec d$ are unit vectors such that
$$\left(\vec a\times\vec b\right)\cdot\left(\vec c\times\vec d\right)=1\quad\text{and}\quad\vec a\cdot\vec c=\frac12,$$
then

Match the statements/expressions given in Column I with the values given in Column II.

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat a,\hat b,\hat c$ such that
$$\hat a\cdot\hat b=\hat b\cdot\hat c=\hat c\cdot\hat a=\frac12.$$
Then, the volume of the parallelopiped is

Let two non-collinear unit vectors $\hat a$ and $\hat b$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow{OP}$ (where $O$ is the origin) is given by $\hat a\cos t+\hat b\sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\overrightarrow{OP}$ and $\hat u$ be the unit vector along $\overrightarrow{OP}$. Then,

The unit vector perpendicular to both $L_1$ and $L_2$ is

The number of distinct real values of $\lambda$, for which the vectors $-\lambda^2\hat i+\hat j+\hat k$, $\hat i-\lambda^2\hat j+\hat k$ and $\hat i+\hat j-\lambda^2\hat k$ are coplanar, is

Let the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$, $\overrightarrow{RS}$, $\overrightarrow{ST}$, $\overrightarrow{TU}$ and $\overrightarrow{UP}$ represent the sides of a regular hexagon.

STATEMENT-1: $\overrightarrow{PQ}\times\left(\overrightarrow{RS}+\overrightarrow{ST}\right)\ne\vec0$.

because

STATEMENT-2: $\overrightarrow{PQ}\times\overrightarrow{RS}=\vec0$ and $\overrightarrow{PQ}\times\overrightarrow{ST}\ne\vec0$.

Let $\vec a,\vec b,\vec c$ be unit vectors such that $\vec a+\vec b+\vec c=\vec0$. Which one of the following is correct?