Vector Algebra — JEE Advanced PYQs

For real numbers $\alpha ,\beta ,\gamma ,\delta$ and $\mu$, consider the matrix

Suppose that …

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 …

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

Let $\vec{w}=\hat{i}+\hat{j}-2\hat{k}$, and $\overrightarrow{\mathrm{u}}$ and $\overrightarrow{\mathrm{v}}$ 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 …

Consider the vectors

For two distinct positive real numbers $\alpha$ and $\beta$, define …

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

then the value of …

Let $\overrightarrow{\mathrm{OP}}=\frac{\alpha -1}{\alpha }\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\overrightarrow{\mathrm{OQ}}=\hat{\mathrm{i}}+\frac{\beta -1}{\beta }\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and …

Let the position vectors of the points $P,Q,R$ and $S$ be $\vec{a}=\hat{i}+2\hat{j}-5\hat{k},\vec{b}=3\hat{i}+6\hat{j}+3\hat{k}$, $\overrightarrow{\mathrm{c}}=\frac{17}{5}\hat{\mathrm{i}}+\frac{16}{5}\hat{\mathrm{j}}+7\hat{\mathrm{k}}$ and …

Let P be the plane $\sqrt{3}\mathrm{x}+2\mathrm{y}+3\mathrm{z}=16$ and let $\mathrm{S}=\{\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}:{\alpha }^2+{\beta }^2+{\gamma }^2=1$ and the distance of $(\alpha ,\beta ,\gamma )$ from the plane P is …

Let $\hat{i},\hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let …

Let $\vec{u},\vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and $\vec{u}\cdot \vec{w}=1,\vec{v}\cdot \vec{w}=1,\vec{w}\cdot \vec{w}=4$
If the volume of the parallelopiped …

Let O be the origin and $\overrightarrow{\mathrm{OA}}=2\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}},\overrightarrow{\mathrm{OB}}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and …

In a triangle PQR, let $\vec{a}=\overrightarrow{QR}$, $\vec{b}=\overrightarrow{RP}$ and $\vec{c}=\overrightarrow{PQ}$. If

then the …

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ}=a\hat{i}+b\hat{j}$ and $\overrightarrow{PS}=a\hat{i}-b\hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be the projection vectors of …

Let $\vec{a}=2\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. Consider a vector $\vec{c}=\alpha \vec{a}+\beta \vec{b},\alpha ,\beta \in \mathbb{R}$. If the projection of $\vec{c}$ on the vector $(\vec{a}+\vec{b})$ is $3\sqrt{2}$, th …

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}\cdot \vec{b}=0$. For some $x,y\in \mathbb{R}$, let $\vec{c}=x\vec{a}+y\vec{b}+(\vec{a}\times \vec{b})$. If $|\vec{c}|=2$ and the vector $\vec{c}$ is inclined at the same angle $\alpha$ to bot …

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 …

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

Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ}$, respectively, of a triangle $PQR$. …

Let $\hat{u}=u_1\hat{i}+u_2\hat{j}+u_3\hat{k}$ be a unit vector in ${\mathbb{R}}^3$ and $\hat{w}=\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+2\hat{k})$. Given that there exists a vector $\vec{v}$ in ${\mathbb{R}}^3$ such that $|\hat{u}\times \vec{v}|=1$ and …

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 …

Let $△PQR$ be a triangle. Let $\vec{a}=\overrightarrow{QR},\vec{b}=\overrightarrow{RP}$ and $\vec{c}=\overrightarrow{PQ}$. If $|\vec{a}|=12,|\vec{b}|=4\sqrt{3}$ and $\vec{b}\cdot \vec{c}=24$, then which of the following is (are) true?

Let $\vec{x},\vec{y}$ and $\vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{\pi }{3}$. If $\vec{a}$ is a nonzero vector perpendicular to $\vec{x}$ and $\vec{y}\times \vec{z}$ and $\vec{b}$ is a nonzero vecto …

Let $\vec{a},\vec{b}$, and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi }{3}$ . 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 va …

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 d …

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

Match List-I with List-II and select the correct answer using the code given below the lists :
| P. | Volume of parallelepiped determined by vectors $\vec{a},\vec{b}$ and $\vec{c}$ is 2. Then the volume of the parallelepiped de …

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

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

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 $\frac{1}{\sqrt{3}}$, 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 $\vec{r}.\vec{a}=0$, then the value of …

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

If $\vec{a}$ and $\vec{b}$ are vectors in space given by $\vec{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\vec{b}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}$, then the value of $(2\vec{a}+\vec{b})\cdot [(\vec{a}\times \vec{b})\times (\vec{a}-2\vec{b})]$ is

Match the statements/expressions given in Column I with the values given in Column II.

Column I
(A) Root(s) of the equation $2{\sin }^2\theta +{\sin }^{2}2\theta =2$
(B) Points of discontinuity of the function …

If $\vec{a},\vec{b},\vec{c}$ and $\vec{d}$ are unit vectors such that

and $\vec{a}\cdot \vec{c}=\frac{1}{2}$,
then

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat{a},\hat{b},\hat{c}$ such that

Then, the volume of the parallelopiped is

Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow{OP}$ (where $O$ is the origin) is given by $\hat{a}\cos t+\hat{b}\sin t$. When $P$ is farthest from ori …

Consider the lines


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

The number of distinct real values of $\lambda$, for which the vectors $-{\lambda }^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-{\lambda }^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-{\lambda }^2\hat{k}$ are coplanar, 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 : …

Let $\vec{a},\vec{b},\vec{c}$ be unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$. Which one of the following is correct?

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k},\vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$. A vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$, is

Let $\vec{A}$ be vector parallel to line of intersection of planes $P_1$ and $P_2$ through origin. $P_1$ is parallel to the vectors $2\hat{j}+3\hat{k}$ and $4\hat{j}-3\hat{k}$ and $P_2$ is parallel to $\hat{j}-\hat{k}$ and $3\hat{i}+3\hat{j}$, then the angle between vecto …