Cross product

If the vector $\vec b=3\hat j+4\hat k$ is written as the sum of a vector $\vec b_1$, parallel to $\vec a=\hat i+\hat j$ and a vector $\vec b_2$, perpendicular to $\vec a$, then $\vec b_1\times\vec b_2$ is equal to :

Let $\hat u=u_1\hat i+u_2\hat j+u_3\hat k$ be a unit vector in $\mathbb R^3$ and $\hat w=\dfrac1{\sqrt6}(\hat i+\hat j+2\hat k)$. Given that there exists a vector $\vec v$ in $\mathbb R^3$ such that $|\hat u\times\vec v|=1$ and $\hat w\cdot(\hat u\times\vec v)=1$. Which of the following statement(s) is(are) correct?

A particle of mass $m$ is moving along the side of a square of side '$a$', with a uniform speed $v$ in the $x$-$y$ plane as shown in the figure :

Which of the following statements is false for the angular momentum $\vec L$ about the origin ?

Let $\Delta PQR$ be a triangle. Let $\vec a=\overrightarrow{QR}$, $\vec b=\overrightarrow{RP}$ and $\vec c=\overrightarrow{PQ}$. If $|\vec a|=12$, $|\vec b|=4\sqrt3$ and $\vec b\cdot\vec c=24$, then which of the following is (are) true?

Let $\vec a$ and $\vec b$ be two unit vectors such that $|\vec a+\vec b|=\sqrt3$. If $\vec c=\vec a+2\vec b+3(\vec a\times\vec b)$, then $2|\vec c|$ is equal to :

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

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 $\vec x=3\hat i-6\hat j-\hat k$, $\vec y=\hat i+4\hat j-3\hat k$ 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 :

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 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 $\vec a$ 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 $\vec a=-\hat i-\hat k$, $\vec b=-\hat i+\hat j$ 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

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

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