Vector Algebra

Match List I with List II and select the correct answer using the code given below the lists.

If $\left[\vec a\times\vec b\ \ \vec b\times\vec c\ \ \vec c\times\vec a\right]=\lambda\left[\vec a\ \vec b\ \vec c\right]^2$ then $\lambda$ is equal to :

If $|\vec a|=2$, $|\vec b|=3$ and $|2\vec a-\vec b|=5$, then $|2\vec a+\vec b|$ equals :

The plane containing the line $\dfrac{x-1}1=\dfrac{y-2}2=\dfrac{z-3}3$ and parallel to the line $\dfrac x1=\dfrac y1=\dfrac z4$ passes through the point :

If $|\vec c|^2=60$ and $\vec c\times\left(\hat i+2\hat j+5\hat k\right)=\vec0$, then a value of $\vec c\cdot\left(-7\hat i+2\hat j+3\hat k\right)$ is :

If $\hat x$, $\hat y$ and $\hat z$ are three unit vectors in three-dimensional space, then the minimum value of
$$|\hat x+\hat y|^2+|\hat y+\hat z|^2+|\hat z+\hat x|^2$$
is :

If $\vec x=3\hat i-6\hat j-\hat k$, $\vec y=\hat i+4\hat j-3\hat k$ and $\vec z=3\hat i-4\hat j-12\hat k$, then the magnitude of the projection of $\vec x\times\vec y$ on $\vec z$ is :

Let $\overrightarrow{PR}=3\hat i+\hat j-2\hat k$ and $\overrightarrow{SQ}=\hat i-3\hat j-4\hat k$ determine diagonals of a parallelogram $PQRS$ and $\overrightarrow{PT}=\hat i+2\hat j+3\hat k$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow{PT}$, $\overrightarrow{PQ}$ and $\overrightarrow{PS}$ is

A line $l$ passing through the origin is perpendicular to the lines
$$l_1:(3+t)\hat i+(-1+2t)\hat j+(4+2t)\hat k,\quad-\infty<t<\infty$$
$$l_2:(3+2s)\hat i+(3+2s)\hat j+(2+s)\hat k,\quad-\infty<s<\infty$$
Then, the coordinate(s) of the point(s) on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of $l$ and $l_1$ is (are)

Consider the set of eight vectors $V=\left\{a\hat i+b\hat j+c\hat k:a,b,c\in\{-1,1\}\right\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is

Two lines $L_1:x=5,\ \dfrac y{3-\alpha}=\dfrac z{-2}$ and $L_2:x=\alpha,\ \dfrac y{-1}=\dfrac z{2-\alpha}$ are coplanar. Then $\alpha$ can take value(s)

Consider the lines $L_1:\dfrac{x-1}2=\dfrac y{-1}=\dfrac{z+3}1$, $L_2:\dfrac{x-4}1=\dfrac{y+3}1=\dfrac{z+3}2$ and the planes $P_1:7x+y+2z=3$, $P_2:3x+5y-6z=4$. Let $ax+by+cz=d$ be the equation of the plane passing through the point of intersection of lines $L_1$ and $L_2$, and perpendicular to planes $P_1$ and $P_2$.

Match List I with List II and select the correct answer using the code given below the lists:

Match List I with List II and select the correct answer using the code given below the lists:

If $\vec a$, $\vec b$ and $\vec c$ are unit vectors satisfying $|\vec a-\vec b|^2+|\vec b-\vec c|^2+|\vec c-\vec a|^2=9$, then $|2\vec a+5\vec b+5\vec c|$ is

If $\vec a$ and $\vec b$ are vectors such that $|\vec a+\vec b|=\sqrt{29}$ and $\vec a\times(2\hat i+3\hat j+4\hat k)=(2\hat i+3\hat j+4\hat k)\times\vec b$, then a possible value of $(\vec a+\vec b)\cdot(-7\hat i+2\hat j+3\hat k)$ is

Let $\vec a=\hat i+\hat j+\hat k$, $\vec b=\hat i-\hat j+\hat k$ and $\vec c=\hat i-\hat j-\hat k$ be three vectors. A vector $\vec v$ in the plane of $\vec a$ and $\vec b$, whose projection on $\vec c$ is $\dfrac1{\sqrt3}$, is given by

The vector(s) which is/are coplanar with vectors $\hat i+\hat j+2\hat k$ and $\hat i+2\hat j+\hat k$, and perpendicular to the vector $\hat i+\hat j+\hat k$ is/are

Let $\vec a=-\hat i-\hat k$, $\vec b=-\hat i+\hat j$ and $\vec c=\hat i+2\hat j+3\hat k$ be three given vectors. If $\vec r$ is a vector such that $\vec r\times\vec b=\vec c\times\vec b$ and $\vec r\cdot\vec a=0$, then the value of $\vec r\cdot\vec b$ is

Match the statements given in Column I with the values given in Column II

Let P, Q, R and S be the points on the plane with position vectors $-2\hat i-\hat j$, $4\hat i$, $3\hat i+3\hat j$ and $-3\hat i+2\hat j$ respectively. The quadrilateral PQRS must be a