Dot product

Let $\vec a,\vec b$ be two vectors, and let $P,Q$ and $R$ be the points with position vectors $\vec a$, $\vec b$ and $\vec a+\vec b$, respectively, with respect to the origin $O$. If $|\vec a+\vec b|=\sqrt{21}$, $|\vec a-\vec b|=3$, and $\vec a$ and $(\vec a-\vec b)$ are perpendicular to each other, then the area of the triangle $OPR$ is

Let $L$ be the straight line joining the points $P(1,2,-1)$ and $Q(2,3,1)$. Let $S$ be the foot of the perpendicular drawn from the point $R(4,-1,5)$ to the line $L$. Another line passing through $R$ intersects $L$ at a point $T$ such that the point $S$ divides the line segment $PT$ internally in the ratio $|PS|:|ST|=1:2$, where $|PS|$ and $|ST|$ are the lengths of the line segments $PS$ and $ST$, respectively.

Then which of the following statements is (are) TRUE ?

If $\vec a$ and $\vec b$ are two vectors such that $|\vec a|=2$ and $|\vec b|=3$, then the maximum value of $3\left|\left(3\vec a+2\vec b\right)\right|+4\left|\left(3\vec a-2\vec b\right)\right|$ is :

Let the vectors $\vec a=-\hat i+\hat j+3\hat k$ and $\vec b=\hat i+3\hat j+\hat k$. For some $\lambda,\mu\in\mathbb R$, let $\vec c=\lambda\vec a+\mu\vec b$. If $\vec c\cdot\left(3\hat i-6\hat j+2\hat k\right)=10$ and $\vec c\cdot\left(\hat i+\hat j+\hat k\right)=-2$, then $|\vec c|^2$ is equal to :

Two adjacent sides of a parallelogram PQRS are given by $\overrightarrow{PQ}=\hat j+\hat k$ and $\overrightarrow{PS}=\hat i-\hat j$. If the side PS is rotated about the point P by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then $\sin^2\left(\dfrac{5\alpha}2\right)-\sin^2\left(\dfrac\alpha2\right)$ is equal to :

Let $\hat u$ and $\hat v$ be unit vectors inclined at an acute angle such that $\left|\hat u\times\hat v\right|=\dfrac{\sqrt3}2$. If $\vec A=\lambda\hat u+\hat v+\left(\hat u\times\hat v\right)$, then $\lambda$ is equal to:

Let $\vec a=\sqrt7\hat i+\hat j-\hat k$ and $\vec b=\hat j+2\hat k$. If $\vec r$ is a vector such that $\vec r\times\vec a+\vec a\times\vec b=\vec0$ and $\vec r\cdot\vec a=0$, then $\left|3\vec r\right|^2$ is equal to:

If $\vec a=\hat i+\hat j+\hat k$, $\vec b=\hat j-\hat k$ and $\vec c$ be three vectors such that $\vec a\times\vec c=\vec b$ and $\vec a\cdot\vec c=3$, then $\vec c\cdot\left(\vec a-2\vec b\right)$ is equal to ________.

Let $\vec a=4\hat i-\hat j+3\hat k$, $\vec b=10\hat i+2\hat j-\hat k$ and a vector $\vec c$ be such that $2\left(\vec a\times\vec b\right)+3\left(\vec b\times\vec c\right)=\vec0$. If $\vec a\cdot\vec c=15$, then $\vec c\cdot\left(\hat i+\hat j-3\hat k\right)$ is equal to :

Let $\vec p = 2\hat i + \hat j + 3\hat k$ and $\vec q = \hat i - \hat j + \hat k$. If for some real numbers $\alpha$, $\beta$, and $\gamma$, we have

$$15\hat i + 10\hat j + 6\hat k = \alpha\left(2\vec p + \vec q\right) + \beta\left(\vec p - 2\vec q\right) + \gamma\left(\vec p \times \vec q\right),$$

then the value of $\gamma$ is _____.

Let $\hat i, \hat j$ and $\hat k$ be the unit vectors along the three positive coordinate axes. Let
$$\vec a = 3\hat i + \hat j - \hat k,\qquad \vec b = \hat i + b_2\hat j + b_3\hat k,\ b_2, b_3 \in \mathbb{R},\qquad \vec c = c_1\hat i + c_2\hat j + c_3\hat k,\ c_1, c_2, c_3 \in \mathbb{R}$$
be three vectors such that $b_2b_3 > 0$, $\vec a\cdot\vec b = 0$ and
$$\begin{pmatrix}0 & -c_3 & c_2\\ c_3 & 0 & -c_1\\ -c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix}1\\ b_2\\ b_3\end{pmatrix} = \begin{pmatrix}3 - c_1\\ 1 - c_2\\ -1 - c_3\end{pmatrix}.$$
Then, which of the following is/are TRUE?

Let $\hat a,\hat b$ be unit vectors. If $\vec c$ be a vector such that the angle between $\hat a$ and $\vec c$ is $\dfrac\pi{12}$, and $\hat b=\vec c+2\left(\vec c\times\hat a\right)$, then $\left|6\vec c\right|^2$ is equal to :

Let $\hat a$ and $\hat b$ be two unit vectors such that $\left|\left(\hat a+\hat b\right)+2\left(\hat a\times\hat b\right)\right|=2$. If $\theta\in(0,\pi)$ is the angle between $\hat a$ and $\hat b$, then among the statements:

(S1) : $2\left|\hat a\times\hat b\right|=\left|\hat a-\hat b\right|$

(S2) : The projection of $\hat a$ on $\left(\hat a+\hat b\right)$ is $\frac12$

Let $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $a_i>0$, $i=1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $\vec a$ on the vector $3\hat i+4\hat j$ be 7. Let $\vec b$ be a vector obtained by rotating $\vec a$ with 90°. If $\vec a$, $\vec b$ and $x$-axis are coplanar, then projection of a vector $\vec b$ on $3\hat i+4\hat j$ is equal to :

Let $\theta$ be the angle between the vectors $\vec a$ and $\vec b$, where $|\vec a|=4$, $|\vec b|=3$ and $\theta\in\left(\dfrac\pi4,\dfrac\pi3\right)$. Then
$$\left|\left(\vec a-\vec b\right)\times\left(\vec a+\vec b\right)\right|^2+4\left(\vec a\cdot\vec b\right)^2$$
is equal to ______.

Let $\vec a=\hat i+\hat j+2\hat k$, $\vec b=2\hat i-3\hat j+\hat k$ and $\vec c=\hat i-\hat j+\hat k$ be three given vectors. Let $\vec v$ be a vector in the plane of $\vec a$ and $\vec b$ whose projection on $\vec c$ is $\dfrac2{\sqrt3}$. If $\vec v\cdot\hat j=7$, then $\vec v\cdot\left(\hat i+\hat k\right)$ is equal to :

Let $\vec a$ and $\vec b$ be the vectors along the diagonals of a parallelogram having area $2\sqrt2$. Let the angle between $\vec a$ and $\vec b$ be acute, $|\vec a|=1$, and $\left|\vec a\cdot\vec b\right|=\left|\vec a\times\vec b\right|$. If $\vec c=2\sqrt2\left(\vec a\times\vec b\right)-2\vec b$, then an angle between $\vec b$ and $\vec c$ is

If $\vec a=2\hat i+\hat j+3\hat k$, $\vec b=3\hat i+3\hat j+\hat k$ and $\vec c=c_1\hat i+c_2\hat j+c_3\hat k$ are coplanar vectors and $\vec a\cdot\vec c=5$, $\vec b\perp\vec c$, then $122\left(c_1+c_2+c_3\right)$ is equal to ______.

Let $\vec a=\alpha\hat i+2\hat j-\hat k$ and $\vec b=-2\hat i+\alpha\hat j+\hat k$, where $\alpha\in\mathbf R$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec a$ and $\vec b$ is $\sqrt{15\left(\alpha^2+4\right)}$, then the value of $2\left|\vec a\right|^2+\left(\vec a\cdot\vec b\right)\left|\vec b\right|^2$ is equal to :

Let $\vec a$ be a vector which is perpendicular to the vector $3\hat i+\dfrac12\hat j+2\hat k$. If $\vec a\times\left(2\hat i+\hat k\right)=2\hat i-13\hat j-4\hat k$, then the projection of the vector $\vec a$ on the vector $2\hat i+2\hat j+\hat k$ is :