Vector Algebra — JEE Main PYQs

For $\lambda >0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{i}+\lambda \hat{j}-3\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. If the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are mutually perpendicular, then the value of …

If $A(1,-1,2),B(5,7,-6),C(3,4,-10)$ and $D(-1,-4,-2)$ are the vertices of a quadrilateral ABCD, then its area is

Let $\vec{a}=\hat{i}-3\hat{j}+7\hat{k},\vec{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+2\vec{b})\times \vec{c}=3(\vec{c}\times \vec{a})$. If $\vec{a}\cdot \vec{c}=130$, then $\vec{b}\cdot \vec{c}$ is equal to ____.

Let $\vec{a}=2\hat{i}+5\hat{j}-\hat{k}$, $\vec{b}=2\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{c}$ be three vectors such that $(\vec{c}+\hat{i})\times (\vec{a}+\vec{b}+\hat{i})=\vec{a}\times (\vec{c}+\hat{i})$. If $\vec{a}\cdot \vec{c}=-29$, then …

Consider three vectors $\vec{a},\vec{b},\vec{c}$. Let $|\vec{a}|=2,|\vec{b}|=3$ and $\vec{a}=\vec{b}\times \vec{c}$. If $\alpha \in \left[0,\frac{\pi }{3}\right]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of …

Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$, $\vec{b}=3\hat{i}+4\hat{j}-5\hat{k}$ and a vector $\vec{c}$ be such that $\vec{a}\times (\vec{b}+\vec{c})+\vec{b}\times \vec{c}=\hat{i}+8\hat{j}+13\hat{k}$. If $\vec{a}\cdot \vec{c}=13$, then $(24-\vec{b}\cdot \vec{c})$ is equal t …

Let $\alpha \beta \gamma =45$; $\alpha ,\beta ,\gamma \in R$. If $x(\alpha ,1,2)+y(1,\beta ,2)+z(2,3,\gamma )=(0,0,0)$ for some $x,y,z\in R$, $xyz\ne 0$, then $6\alpha +4\beta +\gamma$ is equal to ____.

Let $\vec{a}=2\hat{i}+\hat{j}-\hat{k},\vec{b}=((\vec{a}\times (\hat{i}+\hat{j}))\times \hat{i})\times \hat{i}$. Then the square of the projection of $\vec{a}$ on $\vec{b}$ is

Let $\vec{a}=6\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $|\vec{c}|\ge 6,\vec{a}\cdot \vec{c}=6|\vec{c}|,|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$ and …

The set of all $\alpha$, for which the vector $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$ are inclined at an obtuse angle for all $t\in \mathbb{R}$ is :

Let $\vec{a}=9\hat{i}-13\hat{j}+25\hat{k},\vec{b}=3\hat{i}+7\hat{j}-13\hat{k}$ and $\overrightarrow{\mathrm{c}}=17\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be three given vectors. If $\overrightarrow{\mathrm{r}}$ is a vector such that …

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}$ and $\overrightarrow{\mathrm{c}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}$ be three vectors. Let $\overrightarrow{\mathrm{r}}$ be a unit vector along $\vec{b}+\vec{c}$. If …

Let $\vec{a}=4\hat{i}-\hat{j}+\hat{k},\vec{b}=11\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that
$(\vec{a}+\vec{b})\times \vec{c}=\vec{c}\times (-2\vec{a}+3\vec{b})$.
If $(2\vec{a}+3\vec{b})\cdot \vec{c}=1670$, then $|\vec{c}{|}^2$ is equal to :

Let $\overrightarrow{\mathrm{OA}}=2\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{OB}}=6\overrightarrow{\mathrm{a}}+5\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{OC}}=3\overrightarrow{\mathrm{b}}$, where O is the origin. If the area of the parallelogr …

Let three vectors $\vec{a}=\alpha \hat{i}+4\hat{j}+2\hat{k}$, $\overrightarrow{\mathrm{b}}=5\hat{\mathrm{i}}+3\hat{\mathrm{j}}+4\hat{\mathrm{k}},\overrightarrow{\mathrm{c}}=x\hat{\mathrm{i}}+y\hat{\mathrm{j}}+z\hat{\mathrm{k}}$ form a triangle such that …

Between the following two statements:

Statement-I : Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\overrightarrow{\mathrm{b}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$. Then the vector $\overrightarrow{\mathrm{r}}$ satisfying …

Let $\vec{a}=2\hat{i}+\alpha \hat{j}+\hat{k},\dot{b}=-\hat{i}+\hat{k},\vec{c}=\beta \hat{j}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha \beta =-6$. Let the values of the ordered pair ( $\alpha ,\beta$ ) for which the area of the parallelogram of diag …

Let $\overrightarrow{OA}=\vec{a},\overrightarrow{OB}=12\vec{a}+4\vec{b}$ and $\overrightarrow{OC}=\vec{b}$, where $O$ is the origin. If S is the parallelogram with adjacent sides OA and OC, then $\frac{\text{area of the quadrilateral }\mathrm{OABC}}{\text{area of S}}$ …

Let a unit vector $\hat{\mathrm{u}}=x\hat{\mathrm{i}}+y\hat{\mathrm{j}}+z\hat{\mathrm{k}}$ make angles $\frac{\pi }{2},\frac{\pi }{3}$ and $\frac{2\pi }{3}$ with the vectors …

Let $\vec{a}=3\hat{i}+2\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})\times \vec{c}=2(\vec{a}\times \vec{b})+24\hat{j}-6\hat{k}$ and $(\vec{a}-\vec{b}+\hat{i})\cdot \vec{c}=-3$. Then …

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$; $\vec{a}\cdot \vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times \vec{b})-3\vec{b}$, then the angle betwee …

Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram ABCD. If the diagonal $\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the parallelogram is equal to

The position vectors of the vertices $\mathrm{A},\mathrm{B}$ and C of a triangle are $2\hat{i}-3\hat{j}+3\hat{k},2\hat{i}+2\hat{j}+3\hat{k}$ and $-\hat{\mathrm{i}}+\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ respectively. Let $l$ denotes the length of the angle bisector AD …

Let the position vectors of the vertices A , B and C of a triangle be $2\hat{i}+2\hat{j}+\hat{k},\hat{i}+2\hat{j}+2\hat{k}$ and $2\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ respectively. Let $l_1,l_2$ and $l_3$ be the lengths of perpendiculars drawn …

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k},\vec{b}=3(\hat{i}-\hat{j}+\hat{k})$. Let $\vec{c}$ be the vector such that $\vec{a}\times \vec{c}=\vec{b}$ and $\vec{a}\cdot \vec{c}=3$. Then …

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and $\alpha \hat{\mathrm{i}}+2\alpha \hat{\mathrm{j}}-2\hat{\mathrm{k}}$ is acute, is ______.

Consider a $△\mathrm{ABC}$ where $\mathrm{A}(1,3,2),\mathrm{B}(-2,8,0)$ and $\mathrm{C}(3,6,7)$. If the angle bisector of $\angle \mathrm{BAC}$ meets the line BC at D , then the length of the projection of the vector $\overrightarrow{\mathrm{AD}}$ on the vecto …

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\overrightarrow{\mathrm{c}}=4\hat{\mathrm{i}}+{\mathrm{c}}_2\hat{\mathrm{j}}+{\mathrm{c}}_3\hat{\mathrm{k}}$ be three vectors such that …

Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k},\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ be three vectors. If a vectors $\overrightarrow{\mathrm{p}}$ satisfies …

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a}\cdot \vec{b}=2$. If $\vec{c}=(2\vec{a}\times \vec{b})-3\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192{\sin }^2\alpha$ is equal to ____

Let $\vec{a}=-5\hat{i}+\hat{j}-3\hat{k},\vec{b}=\hat{i}+2\hat{j}-4\hat{k}$ and $\vec{c}=((((\vec{a}\times \vec{b})\times \hat{i})\times \hat{i})\times \hat{i})$. Then $\vec{c}\cdot (-\hat{i}+\hat{j}+\hat{k})$ is equal to

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$, $\vec{b}+6\vec{c}$ is collinear with $\vec{a}$ and $\vec{a}+\alpha \vec{b}+\beta \vec{c}=\vec{0}$ …

Let O be the origin and the position vector of A and B be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle \mathrm{AOB}$ meets the line AB at C, then the length of OC is

Let $\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k},\alpha ,\beta \in R$. Let a vector $\vec{b}$ be such that the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is $\frac{\pi }{4}$ and $|\overrightarrow{\mathrm{b}}{|}^2=6$, If …

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\overrightarrow{\mathrm{b}}|=1$ and $|\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{a}}|=2$. Then $|(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{a}})-\overrightarrow{\mathrm{b}}{|}^2$ is …

The sum of all values of $\alpha$, for which the points whose position vectors $\hat{i}-2\hat{j}+3\hat{k}$, $2\hat{i}-3\hat{j}+4\hat{k}$, $(\alpha +1)\hat{i}+2\hat{k}$ and $9\hat{i}+(\alpha -8)\hat{j}+6\hat{k}$ are coplanar, is equal to

Let the vectors $\vec{a},\vec{b},\vec{c}$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by $\vec{a},\vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to

If the points with vectors $\alpha \hat{\mathrm{i}}+10\hat{\mathrm{j}}+13\hat{\mathrm{k}}$, $6\hat{i}+11\hat{j}+11\hat{k}$, $\frac{9}{2}\hat{i}+\beta \hat{j}-8\hat{k}$ are collinear, then $(19\alpha -6\beta {)}^2$ is equal to

Let $\vec{a}=6\hat{i}+9\hat{j}+12\hat{k},\vec{b}=\alpha \hat{i}+11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}\times \vec{c}=\vec{a}\times \vec{b}$. If $\vec{a}\cdot \vec{c}=-12$, …

Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k},\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$ and $\vec{a}\cdot \vec{d}=18$, Then $|\vec{a}\times \vec{d}{|}^2$ is …

Let the position vectors of the points $\mathrm{A},\mathrm{B},\mathrm{C}$ and $D$ be $5\hat{i}+5\hat{j}+2\lambda k,\hat{i}+2\hat{j}+3k,-2\hat{i}+\lambda \hat{j}+4k$ and $-\hat{\mathrm{i}}+5\hat{\mathrm{j}}+6\mathrm{k}$. Let the set …

The area of the quadrilateral ABCD with vertices $\mathrm{A}(2,1,1),\mathrm{B}(1,2,5),\mathrm{C}(-2,-3,5)$ and $\mathrm{D}(1,-6,-7)$ is equal to

Let the vectors ${\overrightarrow{\mathrm{u}}}_1=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{a}\hat{\mathrm{k}}$, ${\overrightarrow{\mathrm{u}}}_2=\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and …

Let $\vec{a}=2\hat{i}+7\hat{j}-\hat{k},\vec{b}=3\hat{i}+5\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c}\cdot \vec{d}=12$. Then …

If the points P and Q are respectively the circumcentre and the orthocentre of a $△\mathrm{ABC}$, then $\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PC}}$ is equal to

Let O be the origin and the position vector of the point P be $-\hat{i}-2\hat{j}+3\hat{k}$. If the position vectors of the points $\mathrm{A},\mathrm{B}$ and C are $-2\hat{i}+\hat{j}-3\hat{k},2\hat{i}+4\hat{j}-2\hat{k}$ and $-4\hat{i}+2\hat{j}-\hat{k}$ respecti …

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $\overrightarrow{OP}=\vec{u}$, $\overrightarrow{OR}=\vec{v}$ and $\overrightarrow{OQ}=\alpha \vec{u}+\beta \vec{v}$, then $\alpha ,{\beta }^2$ are the roots of the equation

Let $a,b,c$ be three distinct real numbers, none equal to one. If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

Let $\lambda \in \mathbb{Z},\vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}+\vec{c})\times \vec{c}=\vec{0},\vec{a}\cdot \vec{c}=-17$ and $\vec{b}\cdot \vec{c}=-20$. Then …

Let $|\vec{a}|=2$, $|\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi }{4}$. Then $|(\vec{a}+2\vec{b})\times (2\vec{a}-3\vec{b}){|}^2$ is equal to