Vector Algebra

Consider the cube in the first octant with sides $OP$, $OQ$ and $OR$ of length 1, along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O(0, 0, 0)$ is the origin. Let $S\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT$. If $\vec{p} = \overrightarrow{SP}$, $\vec{q} = \overrightarrow{SQ}$, $\vec{r} = \overrightarrow{SR}$ and $\vec{t} = \overrightarrow{ST}$, then the value of $|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})|$ is _____ .

The equation of the plane passing through the point $(1, 1, 1)$ and perpendicular to the planes $2x + y - 2z = 5$ and $3x - 6y - 2z = 7$, is

Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point $S$ is such that

$$\overrightarrow{OP}\cdot\overrightarrow{OQ} + \overrightarrow{OR}\cdot\overrightarrow{OS} = \overrightarrow{OR}\cdot\overrightarrow{OP} + \overrightarrow{OQ}\cdot\overrightarrow{OS} = \overrightarrow{OQ}\cdot\overrightarrow{OR} + \overrightarrow{OP}\cdot\overrightarrow{OS}$$

Then the triangle $PQR$ has $S$ as its

$|\overrightarrow{OX} \times \overrightarrow{OY}| =$

Let $\vec a=2\hat i+\hat j-2\hat k$ and $\vec b=\hat i+\hat j$. Let $\vec c$ be a vector such that $|\vec c-\vec a|=3$, $\left|(\vec a\times\vec b)\times\vec c\right|=3$ and the angle between $\vec c$ and $\vec a\times\vec b$ be 30°. Then $\vec a\cdot\vec c$ is equal to :

The line of intersection of the planes $\vec r\cdot(3\hat i-\hat j+\hat k)=1$ and $\vec r\cdot(\hat i+4\hat j-2\hat k)=2$, is :

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat i-6\hat j$ and $3\hat i+4\hat j-12\hat k$, is :

In a certain region static electric and magnetic fields exist. The magnetic field is given by $\vec B=B_0\left(\hat i+2\hat j-4\hat k\right)$. If a test charge moving with a velocity $\vec v=v_0\left(3\hat i-\hat j+2\hat k\right)$ experiences no force in that region, then the electric field in the region, in SI units, is :

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?

Let $\vec a$, $\vec b$ and $\vec c$ be three unit vectors such that $\vec a\times\left(\vec b\times\vec c\right)=\dfrac{\sqrt3}2\left(\vec b+\vec c\right)$. If $\vec b$ is not parallel to $\vec c$, then the angle between $\vec a$ and $\vec b$ is :

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 ?

In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively $3\hat i+\hat j-\hat k$, $-\hat i+3\hat j+p\hat k$ and $5\hat i+q\hat j-4\hat k$, then the point (p, q) lies on a line :

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?

Suppose that $\vec p$, $\vec q$ and $\vec r$ are three non-coplanar vectors in $\mathbb R^3$. Let the components of a vector $\vec s$ along $\vec p$, $\vec q$ and $\vec r$ be 4, 3 and 5, respectively. If the components of this vector $\vec s$ along $(-\vec p+\vec q+\vec r)$, $(\vec p-\vec q+\vec r)$ and $(-\vec p-\vec q+\vec r)$ are $x$, $y$ and $z$, respectively, then the value of $2x+y+z$ is

Let $\vec a$, $\vec b$ and $\vec c$ be three non-zero vectors such that no two of them are collinear and $(\vec a\times\vec b)\times\vec c=\dfrac13|\vec b||\vec c|\vec a$. If θ is the angle between vectors $\vec b$ and $\vec c$, then a value of $\sin\theta$ is :

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 :

In a parallelogram ABCD, $|\vec{AB}|=a$, $|\vec{AD}|=b$ and $|\vec{AC}|=c$, then $\vec{DB}\cdot\vec{AB}$ has the value :

Let $\vec x$, $\vec y$ and $\vec z$ be three vectors each of magnitude $\sqrt2$ and the angle between each pair of them is $\frac\pi3$. If $\vec a$ is a nonzero vector perpendicular to $\vec x$ and $\vec y\times\vec z$ and $\vec b$ is a nonzero vector perpendicular to $\vec y$ and $\vec z\times\vec x$, then

Let $\vec a$, $\vec b$ and $\vec c$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac\pi3$. If $\vec a\times\vec b+\vec b\times\vec c=p\vec a+q\vec b+r\vec c$, where $p$, $q$ and $r$ are scalars, then the value of
$$\frac{p^2+2q^2+r^2}{q^2}$$
is