Vector Algebra — JEE Main PYQs

Let $\vec{a}=-\hat{i}+2\hat{j}+2\hat{k},\vec{b}=8\hat{i}+7\hat{j}-3\hat{k}$ and $\vec{c}$ be a vector such that $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$. If …

Let $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$ be vectors such that $|\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|=\sqrt{29}$ and …

For a triangle $ABC$, let $\vec{p}=\overrightarrow{BC},\vec{q}=\overrightarrow{CA}$ and $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{BA}}$. If $|\overrightarrow{\mathrm{p}}|=2\sqrt{3},|\overrightarrow{\mathrm{q}}|=2$ and $\cos \theta =\frac{1}{\sqrt{3}}$, where $\theta$ …

Let $\overrightarrow{AB}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\lambda \hat{k},\lambda \in \mathbb{R}$. Let the projection of the vector …

Let $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda \hat{j}+2\hat{k}$, $\lambda \in \mathbb{Z}$ be two vectors. Let $\vec{c}=\vec{a}\times \vec{b}$ and $\vec{d}$ be a vector of magnitude 2 in yz-plane. If $|\vec{c}|=\sqrt{53}$, then the maximum p …

Let a vector $\vec{a}=\sqrt{2}\hat{i}-\hat{j}+\lambda \hat{k},\lambda >0$, make an obtuse angle with the vector $\vec{b}=-{\lambda }^2\hat{i}+4\sqrt{2}\hat{j}+4\sqrt{2}\hat{k}$ and an angle $\theta ,\frac{\pi }{6}<\theta <\frac{\pi }{2}$, with the positive $z$-axis. If …

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

Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k},\vec{b}=2\hat{i}+\hat{j}-\hat{k},\vec{c}=\lambda \hat{i}+\hat{j}+\hat{k}$ and $\vec{v}=\vec{a}\times \vec{b}$. If $\vec{v}\cdot \vec{c}=11$ and the length of the projection of $\vec{b}$ on $\vec{c}$ is $p$, then …

Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=2(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}})$. If …

Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k},\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\vec{a}\times \vec{b}$. Let $\overrightarrow{\mathrm{d}}$ be a vector such that …

Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that
$2(\vec{a}\times \vec{c})+3(\vec{b}\times \vec{c})=\overrightarrow{0}$ and $(\vec{a}-\vec{b})\cdot \vec{c}=-97$, then …

For three unit vectors $\vec{a},\vec{b},\vec{c}$ satisfying
$|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}{|}^2+|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}{|}^2+|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}{|}^2=9$ and …

Let PQR be a triangle such that $\overrightarrow{\mathrm{PQ}}=-2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{PR}}=a\hat{\mathrm{i}}+b\hat{\mathrm{j}}-4\hat{\mathrm{k}},a,b\in \mathbb{Z}$. Let $S$ be the point on QR, which is equidistan …

Let P be a point in the plane of the vector $\overrightarrow{\mathrm{AB}}=3\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{AC}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ such that P is equidistant from the lines AB and AC . If …

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=2$ and $|\vec{b}|=3$, then the maximum value of $3|(3\vec{a}+2\vec{b})|+4|(3\vec{a}-2\vec{b})|$ is :

Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{\mathrm{u}}\times \hat{\mathrm{v}}|=\frac{\sqrt{3}}{2}$. If

Then $\lambda$ …

Let ${\vec{a}}_k=\left(\tan {\theta }_k\right)\hat{i}+\hat{j}$ and ${\vec{b}}_k=\hat{i}-\left(\cos {\theta }_k\right)\hat{j}$, where ${\theta }_k=\frac{2^{k-1}\pi }{2^n+1}$, for some $n\in \mathbb{N},n>5$. Then the value of …

Let $\vec{a}=\sqrt{7}\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+2\hat{k}$. If $\vec{r}$ is a vector such that $\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=\overrightarrow{0}$ and …

Let O be the origin, $\overrightarrow{\mathrm{OP}}=\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{OQ}}=\overrightarrow{\mathrm{b}}$. If $R$ is the point on $\overrightarrow{\mathrm{OP}}$ such that $\overrightarrow{\mathrm{OP}}=5\overrightarrow{\mathrm{OR}}$, …

Let $\vec{a}=2\hat{i}+3\hat{j}+3\hat{k}$ and $\vec{b}=6\hat{i}+3\hat{j}+3\hat{k}$. Then the square of the area of the triangle with adjacent sides determined by the vectors $(2\vec{a}+3\vec{b})$ and $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})$ is :

If $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{a}\times \vec{c}=\vec{b}$ and $\vec{a}\cdot \vec{c}=3$, then $\vec{c}\cdot (\vec{a}-2\vec{b})$ is equal to

Let $\vec{a}=4\hat{i}-\hat{j}+3\hat{k},\vec{b}=10\hat{i}+2\hat{j}-\hat{k}$ and a vector $\vec{c}$ be such that $2(\vec{a}\times \vec{b})+3(\vec{b}\times \vec{c})=\overrightarrow{0}$. If $\vec{a}\cdot \vec{c}=15$, then …

Let the vectors $\vec{a}=-\hat{i}+\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+\hat{k}$. For some $\lambda ,\mu \in \mathbf{R}$, let $\vec{c}=\lambda \vec{a}+\mu \vec{b}$. If $\vec{c}\cdot (3\hat{i}-6\hat{j}+2\hat{k})=10$ and …

Two adjacent sides of a parallelogram PQRS are given by $\overrightarrow{\mathrm{PQ}}=\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{PS}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}$. If the side PS is rotated about the point P by an acute angle $\alpha$ in the plane …

Let $\overrightarrow{\mathrm{c}}$ be the projection vector of $\overrightarrow{\mathrm{b}}=\lambda \hat{\mathrm{i}}+4\hat{\mathrm{k}},\lambda >0$, on the vector $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vect …

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{3}$. If $\lambda \vec{a}+2\vec{b}$ and $3\vec{a}-\lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is :

Let the arc AC of a circle subtend a right angle at the centre O . If the point B on the $\mathrm{arc}\mathrm{AC}$, divides the arc AC such that $\frac{\text{ length of }\mathrm{arc}\mathrm{AB}}{\text{ length of }\mathrm{arc}\mathrm{BC}}=\frac{1}{5}$, and …

Let the position vectors of the vertices $\mathrm{A},\mathrm{B}$ and C of a tetrahedron ABCD be $\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}},\hat{\mathrm{i}}+3\hat{\mathrm{j}}-2\hat{\mathrm{k}}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively. The altitude from the vert …

Let the point A divide the line segment joining the points $\mathrm{P}(-1,-1,2)$ and $\mathrm{Q}(5,5,10)$ internally in the ratio $r:1(r>0)$. If $O$ is the origin and …

Let â be a unit vector perpendicular to the vectors $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-\hat{\mathrm{k}}$, and makes an angle of …

Let $\vec{a}=2\hat{i}-\hat{j}+3\hat{k},\vec{b}=3\hat{i}-5\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times \vec{c}=\vec{c}\times \vec{b}$ and …

Let $\mathrm{A},\mathrm{B},\mathrm{C}$ be three points in xy -plane, whose position vector are given by $\sqrt{3}\hat{i}+\hat{j},\hat{i}+\sqrt{3}\hat{j}$ and $a\hat{\mathrm{i}}+(1-\mathrm{a})\hat{\mathrm{j}}$ respectively with respect to the origin O . If the distance of t …

If the components of $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}$ along and perpendicular to $\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ respectively, are $\frac{16}{11}(3\hat{i}+\hat{j}-\hat{k})$ and …

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=2\hat{i}+2\hat{j}+\hat{k}$ and $\vec{d}=\vec{a}\times \vec{b}$. If $\overrightarrow{\mathrm{c}}$ is a vector such that …

Let the position vectors of three vertices of a triangle be $4\vec{p}+\vec{q}-3\vec{r},-5\vec{p}+\vec{q}+2\vec{r}$ and $2\vec{p}-\vec{q}+2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are …

Let $\vec{a}=3\hat{i}-\hat{j}+2\hat{k},\vec{b}=\vec{a}\times (\hat{i}-2\hat{k})$ and $\vec{c}=\vec{b}\times \hat{k}$.

Then the projection of $\vec{c}-2\hat{j}$ on $\vec{a}$ is:

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\overrightarrow{\mathrm{c}}$ is coplanar with $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$. If the vector $\overrightarrow{\mathrm{c}}$ is per …

Let $\vec{a}=2\hat{i}-3\hat{j}+\hat{k},\vec{b}=3\hat{i}+2\hat{j}+5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a}-\vec{c})\times \vec{b}=-18\hat{i}-3\hat{j}+12\hat{k}$ and $\vec{a}\cdot \vec{c}=3$. If $\vec{b}\times \vec{c}=\vec{d}$, then …

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k},\vec{b}=3\hat{i}-3\hat{j}+3\hat{k}$, $\overrightarrow{\mathrm{c}}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}}$ be a vector such that $\vec{b}\times \vec{d}=\vec{c}\times \vec{d}$ and …

If $\vec{a}$ is nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is:

Let ABCD be a tetrahedron such that the edges $\mathrm{AB},\mathrm{AC}$ and AD are mutually perpendicular. Let the areas of the triangles $\mathrm{ABC},\mathrm{ACD}$ and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of the …

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=3\hat{i}+2\hat{j}-\hat{k},\vec{c}=\lambda \hat{j}+\mu \hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a}\times \hat{d}=\vec{b}\times \hat{d}$ and $\vec{c}\cdot \hat{d}=1$, If $\overrightarrow{\mathrm{c}}$ is pe …

Let the three sides of a triangle ABC be given by the vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let G be the centroid of the triangle ABC. Then …

Consider two vectors $\overrightarrow{\mathrm{u}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{v}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}},\lambda >0$. The angle between them is given by …

Let $\vec{a}$ and $\vec{b}$ be the vectors of the same magnitude such that $\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1$. Then $\frac{|\vec{a}+\vec{b}{|}^2}{|\vec{a}{|}^2}$ is :

Let the angle $\theta$, $0<\theta <\frac{\pi }{2}$ between two unit vectors $\hat{a}$ and $\hat{b}$ be ${\sin }^{-1}\left(\frac{\sqrt{65}}{9}\right)$. If the vector $\vec{c}=3\hat{a}+6\hat{b}+9(\hat{a}\times \hat{b})$, then the value of …

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is

Let a unit vector which makes an angle of $60^{\circ }$ with $2\hat{i}+2\hat{j}-\hat{k}$ and an angle of $45^{\circ }$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then $\vec{C}+\left(-\frac{1}{2}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k}\right)$ is

Let ABC be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\overrightarrow{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and $\overrightarrow{AC}=6\hat{i}+d\hat{j}-2\hat{k}$, $d>0$. Then the square of the length of the largest side of …

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\vec{c}=x\hat{i}+2\hat{j}+3\hat{k}$, $x\in \mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a}\cdot \vec{d}=1$, then …