Vector Algebra — JEE Main PYQs

Let $\vec{a}=\alpha \hat{i}+3\hat{j}-\hat{k},\vec{b}=3\hat{i}-\beta \hat{j}+4\hat{k}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-2\hat{\mathrm{k}}$ where $\alpha ,\beta \in \mathrm{R}$, be three vectors. If the projection of $\vec{a}$ on …

If $\vec{a}\cdot \vec{b}=1,\vec{b}\cdot \vec{c}=2$ and $\vec{c}\cdot \vec{a}=3$, then the value of $[\vec{a}\times (\vec{b}\times \vec{c}),\vec{b}\times (\vec{c}\times \vec{a}),\vec{c}\times (\vec{b}\times \vec{a})]$ is:

Let $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+2\hat{k}$. Then the number of vectors $\vec{b}$ such that $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{b}|\in \{1,2,\dots ,10\}$ is :

Let $\vec{a}=3\hat{i}+\hat{j}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. Let $\vec{c}$ be a vector satisfying $\vec{a}\times (\vec{b}\times \vec{c})=\vec{b}+\lambda \vec{c}$. If $\vec{b}$ and $\vec{c}$ are non-parallel, then the value of $\lambda$ is :

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2\hat{b}+2(\hat{a}\times \hat{b}))$, then the value of $164{\cos }^2\theta$ is equ …

Let $\hat{a},\hat{b}$ be unit vectors. If $\vec{c}$ be a vector such that the angle between $\hat{\mathrm{a}}$ and $\overrightarrow{\mathrm{c}}$ is $\frac{\pi }{12}$, and …

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a}\times \hat{b})|=2$. If $\theta \in (0,\pi )$ is the angle between $\hat{a}$ and $\hat{b}$, then among the statements:
(S1) : $2|\hat{a}\times \hat{b}|=|\hat{a}-\hat{b}|$
(S2) : The projection …

Let $\vec{b}=\hat{i}+\hat{j}+\lambda \hat{k},\lambda \in R$. If $\vec{a}$ is a vector such that $\vec{a}\times \vec{b}=13\hat{i}-\hat{j}-4\hat{k}$ and $\vec{a}\cdot \vec{b}+21=0$, then $(\vec{b}-\vec{a})\cdot (\hat{k}-\hat{j})+(\vec{b}+\vec{a})\cdot (\hat{i}-\hat{k})$ is equ …

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ $a_i>0,i=1,2,3$ be a vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of $\overrightarrow{\mathrm{a}}$ on the vector $3\hat{\mathrm{i}}+4\hat{\mathrm{j}}$ be 7. Let …

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

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

Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$, $\vec{a}\cdot \vec{c}=7$ and $\vec{b}$ is perpendicular to $\vec{c}$, where $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{k}$, then the value of …

If the projection of the vector $\hat{i}+2\hat{j}+\hat{k}$ on the sum of the two vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $-\lambda \hat{i}+2\hat{j}+3\hat{k}$ is 1, then $\lambda$ is equal to ____.

Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $(\vec{a}\times \vec{b})$ and $\vec{c}$ is $\frac{\pi }{6}$, then the val …

Let $\vec{a},\vec{b},\vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36{\cos }^{2}2\theta$ is equal to ____

Let P be a plane passing through the points $(1,0,1)$, $(1,-2,1)$ and $(0,1,-2)$. Let a vector $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}$ be such that $\overrightarrow{\mathrm{a}}$ is parallel to the plane P, per …

In a triangle ABC , if $|\overrightarrow{\mathrm{BC}}|=3,|\overrightarrow{\mathrm{CA}}|=5$ and $|\overrightarrow{\mathrm{BA}}|=7$, then the projection of the vector $\overrightarrow{\mathrm{BA}}$ on $\overrightarrow{\mathrm{BC}}$ is equal to :

For $p>0$, a vector ${\vec{v}}_2=2\hat{i}+(p+1)\hat{j}$ is obtained by rotating the vector ${\vec{v}}_1=\sqrt{3}p\hat{i}+\hat{j}$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta =\frac{(\alpha \sqrt{3}-2)}{(4\sqrt{3}+3)}$, then the value of …

Let $\vec{a}=\hat{i}+2\hat{j}-\hat{k},\vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=\vec{c}\times \vec{a}$ and $\vec{r}\cdot \vec{b}=0$, then $\vec{r}\cdot \vec{a}$ is equal t …

Let a vector $\vec{a}$ be coplanar with vectors $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to $\vec{d}=3\hat{i}+2\hat{j}+6\hat{k}$ and $|\vec{a}|=\sqrt{10}$. Then a possible value of …

Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{a}\times \vec{b}=\vec{c}$, $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{a}|=2$. Then which one of the following is not true?

If $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are perpendicular, then $\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})))$ is equal t …

Let $\vec{a}=\hat{i}+5\hat{j}+\alpha \hat{k},\vec{b}=\hat{i}+3\hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that, $|\vec{b}\times \vec{c}|=5\sqrt{3}$ and $\vec{a}$ is perpendicular to $\vec{b}$. Then the greatest among the …

If vectors ${\vec{a}}_1=x\hat{i}-\hat{j}+\hat{k}$ and ${\vec{a}}_2=\hat{i}+y\hat{j}+z\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i}+y\hat{j}+z\hat{k}$ is :

Let $\vec{a},\vec{b},\vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies …

Let and be two vectors such that and the angle between and is . If is a unit vector, then is equal to:

Let $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=-\hat{i}+2\hat{j}+3\hat{k}$. Then the vector product $(\vec{a}+\vec{b})\times ((\vec{a}\times ((\vec{a}-\vec{b})\times \vec{b}))\times \vec{b})$ is equal to :

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{a}\cdot \vec{b}=1$. If the length of projection vector of the vector $\vec{b}$ on the vector $\vec{a}\times \vec{c}$ is $\ell$, the …

Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\vec{b}=7\hat{i}+\hat{j}-6\hat{k}$.
If $\vec{r}\times \vec{a}=\vec{r}\times \vec{b},\vec{r}\cdot (\hat{i}+2\hat{j}+\hat{k})=-3$, then $\vec{r}\cdot (2\hat{i}-3\hat{j}+\hat{k})$ is equal to :

If $\vec{a}=\alpha \hat{i}+\beta \hat{j}+3\hat{k}$,
$\overrightarrow{\mathrm{b}}=-\beta \hat{\mathrm{i}}-\alpha \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and
$\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}-\hat{\mathrm{k}}$
such that $\vec{a}\cdot \vec{b}=1$ a …

Let the vectors
$(2+a+b)\hat{i}+(a+2b+c)\hat{j}-(b+c)\hat{k}$,
$(1+b)\hat{i}+2bj-b\hat{k}$ and $(2+b)\hat{i}+2bj+(1-b)\hat{k},a,b,c\in R$ be co-planar. Then which of the following is true?

Let $\vec{p}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. If a vector : $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to each of the vectors ( $\vec{p}+\vec{q}$ ) and ( $\vec{p}-\vec{q}$ ), and …

Let $a,b$ and $c$ be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are coplanar, then c is equal to :

If $|\vec{a}|=2,|\vec{b}|=5$ and $|\vec{a}\times \vec{b}|=8$, then $|\vec{a}\cdot \vec{b}|$ is equal to :

If $(\vec{a}+3\vec{b})$ is perpendicular to $(7\vec{a}-5\vec{b})$ and $(\vec{a}-4\vec{b})$ is perpendicular to $(7\vec{a}-2\vec{b})$, then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is ____

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b}\times (\vec{b}\times \vec{c})$. If magnitudes of the vectors $\vec{a},\vec{b}$ and $\vec{c}$ are $\sqrt{2},1$ and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is …

Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k},\vec{b}=3\hat{i}+\beta \hat{j}-\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a}\cdot \vec{b}=-1$ and $\vec{b}\cdot \vec{c}=10$, then …

Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\vec{a}$ is 19 units, the …

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha ,\beta ),(0,\beta )$ …

Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}-3\hat{j}+5\hat{k}$.
If $\vec{r}\times \vec{a}=\vec{b}\times \vec{r},\vec{r}.(a\hat{i}+2\hat{j}+\hat{k})=3$ and $\vec{r}.(2\hat{i}+5\hat{j}-\alpha \hat{k})=-1,\alpha \in R$, then the value of …

Let $\vec{c}$ be a vector perpendicular to the vectors $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. If $\vec{c}\cdot (\hat{i}+\hat{j}+3\hat{k})=8$ then the value of $\vec{c}\cdot (\vec{a}\times \vec{b})$ is equal to ______

Let $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are two non-zero vectors perpendicular to each other and $|\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}}|$. If …

In a triangle ABC , if $|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7,|\overrightarrow{\mathrm{AB}}|=10$, then the projection of the vector $\overrightarrow{\mathrm{AB}}$ on $\overrightarrow{\mathrm{AC}}$ is equal to :

Let $O$ be the origin. Let $\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}$ and $\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k},x,y\in R,x>0$, be such that $|\overrightarrow{PQ}|=\sqrt{20}$ and the vector $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OQ}$. I …

Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. If the vector $\vec{x}$ is perpendicular to $(3\hat{i}+2\hat{j}-\hat{k})$ and its projection on $\vec{a}$ is $\frac{17\sqrt{6}}{2}$, then the …

A vector $\vec{a}$ has components $3p$ and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $\vec{a}$ has components $p+1$ and $\sqrt{10}$, …

A vector $\vec{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}(\alpha ,\beta \in R)$ lies in the plane of the vectors, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4\hat{k}$. If $\vec{a}$ bisects the angle between $\vec{b}$ and $\vec{c}$, then:

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$. If $\lambda =\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ and …

Let the volume of a parallelepiped whose coterminous edges are given by $\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}},\overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ and …

Let $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b}\times \vec{c}=\vec{b}\times \vec{a}$ and $\vec{c}\cdot \vec{a}=0$, then $\vec{c}\cdot \vec{b}$ is equal to: