Vector Algebra

Let $P$ be the plane such that it contains the straight line $\dfrac{x-1}2=\dfrac{y-3}3=\dfrac{z+2}1$ and is perpendicular to the plane $x+2y+3z=4$. Let $P_1$ be the plane which passes through the point $(4,2,2)$ and is parallel to $P$.

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

For real numbers $\alpha,\ \beta,\ \gamma,\ \delta$ and $\mu$, consider the matrix
$$M=\begin{bmatrix}\alpha&\frac1{\sqrt2}&-\frac1{\sqrt2}\\\frac1{\sqrt3}&\beta&\frac1{\sqrt3}\\\gamma&\delta&\mu\end{bmatrix}.$$
Suppose that $MM^T=I$, where $M^T$ is the transpose of the matrix $M$, and $I$ is the $3\times3$ identity matrix. Let
$$\vec u=\alpha\,\hat i+\frac1{\sqrt3}\hat j+\gamma\,\hat k,\qquad\vec v=\frac1{\sqrt2}\hat i+\beta\hat j+\delta\hat k\quad\text{and}\quad\vec w=-\frac1{\sqrt2}\hat i+\frac1{\sqrt3}\hat j+\mu\hat k.$$
Match each entry in List-I to the correct entry in List-II and choose the correct option.

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 a line $L$ passing through the point $(1,1,1)$ be perpendicular to both the vectors $2\hat i+2\hat j+\hat k$ and $\hat i+2\hat j+2\hat k$. If $P(a,b,c)$ is the foot of perpendicular from the origin on the line $L$, then the value of $34(a+b+c)$ 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 $\vec{a_k}=(\tan\theta_k)\hat i+\hat j$ and $\vec{b_k}=\hat i-(\cot\theta_k)\hat j$, where $\theta_k=\dfrac{2^{k-1}\pi}{2^n+1}$, for some $n\in\mathbb N$, $n>5$. Then the value of $\dfrac{\sum_{k=1}^n\left|\vec{a_k}\right|^2}{\sum_{k=1}^n\left|\vec{b_k}\right|^2}$ is ________.

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:

Let O be the origin, $\overrightarrow{OP}=\vec a$ and $\overrightarrow{OQ}=\vec b$. If R is the point on $\overrightarrow{OP}$ such that $\overrightarrow{OP}=5\overrightarrow{OR}$, and M is the point such that $\overrightarrow{OQ}=5\overrightarrow{RM}$, then $\overrightarrow{PM}$ is equal to :

Let a line L be perpendicular to both the lines $L_1:\dfrac{x+1}3=\dfrac{y+3}5=\dfrac{z+5}7$ and $L_2:\dfrac{x-2}1=\dfrac{y-4}4=\dfrac{z-6}7$. If $\theta$ is the acute angle between the lines L and $L_3:\dfrac{x-\frac87}2=\dfrac{y-\frac47}1=\dfrac z2$, then $\tan\theta$ 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 $\vec a=2\hat i+3\hat j+3\hat k$ and $\vec b=6\hat i+3\hat j+3\hat k$. Then the square of the area of the triangle with adjacent sides determined by the vectors $\left(2\vec a+3\vec b\right)$ and $\left(\vec a-\vec b\right)$ is :

Let the image of the point $P(0,-5,0)$ in the line $\dfrac{x-1}2=\dfrac y1=\dfrac{z+1}{-2}$ be the point R and the image of the point $Q\left(0,\dfrac{-1}2,0\right)$ in the line $\dfrac{x-1}{-1}=\dfrac{y+9}4=\dfrac{z+1}1$ be the point S. Then the square of the area of the parallelogram PQRS is ________.

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 a line $L_1$ pass through the origin and be perpendicular to the lines
$L_2:\vec r=(3+t)\hat i+(2t-1)\hat j+(2t+4)\hat k$ and
$L_3:\vec r=(3+2s)\hat i+(3+2s)\hat j+(2+s)\hat k$, $t,s\in\mathbf R$.
If $(a,b,c)$, $a\in\mathbf Z$, is the point on $L_3$ at a distance of $\sqrt{17}$ from the point of intersection of $L_1$ and $L_2$, then $(a+b+c)^2$ is equal to ________.

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec 0$ denote the zero vector. Let $P$, $Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\Delta PQR$ such that

$$\overrightarrow{SP} + 5\,\overrightarrow{SQ} + 6\,\overrightarrow{SR} = \vec 0.$$

Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of

$$\frac{\text{length of the line segment } EF}{\text{length of the line segment } ES}$$

is _____.

Let $\vec w = \hat i + \hat j - 2\hat k$, and $\vec u$ and $\vec v$ be two vectors, such that $\vec u \times \vec v = \vec w$ and $\vec v \times \vec w = \vec u$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that

$$\vec u = \alpha\hat i + \beta\hat j + \gamma\hat k,\quad -t\alpha + \beta + \gamma = 0,\quad \alpha - t\beta + \gamma = 0,\quad\text{and}\quad \alpha + \beta - t\gamma = 0.$$

Match each entry in List-I to the correct entry in List-II and choose the correct option.