Vector Algebra — JEE Main PYQs

Let for a triangle ABC,
$\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}$
$\overrightarrow{CB}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$
$\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta \hat{k}$
If $\delta >0$ and the area of the triangle ABC is $5\sqrt{6}$, then …

Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{d}\cdot \vec{a}=24$, then $|\vec{d}{|}^2$ is equal to

Let $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+3\hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b}\times \vec{c}$ and $|\vec{b}{|}^2=50$, then $\left|72-|\vec{b}+\vec{c}{|}^2\right|$ is equal to _____.

Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and $(\overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{BC}})+(\overrightarrow{\mathrm{AD}}-\overrightarrow{\mathrm{DC}})=\mathrm{k}\overrightarrow{\mathrm{FE}}$, then k …

Let S be the set of all $(\lambda ,\mu )$ for which the vectors $\lambda \hat{i}-\hat{j}+\hat{k},\hat{i}+2\hat{j}+\mu \hat{k}$ and $3\hat{i}-4\hat{j}+5\hat{k}$, where $\lambda -\mu =5$, are coplanar, then …

If four distinct points with position vectors $\vec{a},\vec{b},\vec{c}$ and $\overrightarrow{\mathrm{d}}$ are coplanar; then $[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}]$ is equal to

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=11$, $\vec{b}\cdot (\vec{a}\times \vec{c})=27$ and $\vec{b}\cdot \vec{c}=-\sqrt{3}|\vec{b}|$, then …

For any vector $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, with $|a_i|<1,i=1,2,3$, consider the following statements:
(A): $\max \{|a_1|,|a_2|,|a_3|\}\le |\vec{a}|$
(B): $|\vec{a}|\le 3\max \{|a_1|,|a_2|,|a_3|\}$

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

Let PQR be a triangle. The points $\mathrm{A},\mathrm{B}$ and C are on the sides $\mathrm{QR},\mathrm{RP}$ and PQ respectively such that $\frac{\mathrm{QA}}{\mathrm{AR}}=\frac{\mathrm{RB}}{\mathrm{BP}}=\frac{\mathrm{PC}}{\mathrm{CQ}}=\frac{1}{2}$. Then …

If $\vec{a},\vec{b},\vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a}=\alpha \vec{b}-\hat{n}$, $(\alpha \ne 0)$ and $\vec{b}\cdot \vec{c}=12$, then $|\vec{c}\times (\vec{a}\times \vec{b})|$ is equal …

Let $\vec{a}=2\hat{i}-7\hat{j}+5\hat{k}$, $\vec{b}=\hat{i}+\hat{k}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}$ be three given vectors. If $\overrightarrow{\mathrm{r}}$ is a vector such that …

If $\vec{a}=\hat{i}+2\hat{k},\vec{b}=\hat{i}+\hat{j}+\hat{k},\vec{c}=7\hat{i}-3\hat{j}+4\hat{k}$,
$\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}=\overrightarrow{0}$ and …

Let $\vec{a}=4\hat{i}+3\hat{j}$ and $\vec{b}=3\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}$ is a vector such that $\vec{c}\cdot (\vec{a}\times \vec{b})+25=0$, $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=4$ and projection of $\vec{c}$ on $\vec{a}$ is 1, then the projection of …

If the vectors $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=-2\hat{i}+4\hat{j}-2\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}+\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible value …

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points A, $B,C$ and $D$ be $\vec{a}-\vec{b}+\vec{c},\lambda \vec{a}-3\vec{b}+4\vec{c}$, $-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectivel …

If the four points, whose position vectors are $3\hat{i}-4\hat{j}+2\hat{k},\hat{i}+2\hat{j}-\hat{k},-2\hat{i}-\hat{j}+3\hat{k}$ and $5\hat{i}-2\alpha \hat{j}+4\hat{k}$ are coplanar, then $\alpha$ is equal to

Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k},\vec{a}\cdot \vec{b}=1$ and $\vec{a}\times \vec{b}=\hat{i}-\hat{j}$. Then $\overrightarrow{\mathrm{a}}-6\overrightarrow{\mathrm{b}}$ is equal to

Let $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. Let ${\vec{\beta }}_1$ be parallel to $\vec{\alpha }$ and ${\vec{\beta }}_2$ be perpendicular to $\vec{\alpha }$. If $\vec{\beta }={\vec{\beta }}_1+{\vec{\beta }}_2$, then the value of …

Let $a=\hat{i}+2\hat{j}+\lambda \hat{k},b=3\hat{i}-5\hat{j}-\lambda \hat{k},\vec{a}\cdot \vec{c}=7$, $2\vec{b}\cdot \vec{c}+43=0,\vec{a}\times \vec{c}=\vec{b}\times \vec{c}$. Then $|\vec{a}\cdot \vec{b}|$ is equal to

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

Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $|\overrightarrow{\mathrm{a}}|=\sqrt{31},4|\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{c}}|=2$ and …

A vector v in the first octant is inclined to the x-axis at $60^{\circ }$, to the y-axis at $45^{\circ }$ and to the z-axis at an acute angle. If a plane passing through the points $(\sqrt{2},-1,1)$ and $(\mathrm{a},\mathrm{b},\mathrm{c})$, is normal to …

Let $\lambda \in \mathbb{R},\vec{a}=\lambda \hat{i}+2\hat{j}-3\hat{k},\vec{b}=\hat{i}-\lambda \hat{j}+2\hat{k}$.

If $((\vec{a}+\vec{b})\times (\vec{a}\times \vec{b}))\times (\vec{a}-\vec{b})=8\hat{i}-40\hat{j}-24\hat{k}$, then …

Let $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ be two vectors. Let $|\overrightarrow{\mathrm{a}}|=1,|\overrightarrow{\mathrm{b}}|=4$ and $\vec{a}\cdot \vec{b}=2$. If $\vec{c}=(2\vec{a}\times \vec{b})-3\vec{b}$, then the value of …

Let $\vec{a}=2\hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\overrightarrow{\mathrm{b}}.\overrightarrow{\mathrm{c}}=0$. Consider the following two statement:
(A) …

Let $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ be two vector such that $|\overrightarrow{\mathrm{a}}|=\sqrt{14}$, $|\overrightarrow{\mathrm{b}}|=\sqrt{6}$ and $|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|=\sqrt{48}$. Then …

$\mathrm{A}(2,6,2),\mathrm{B}(-4,0,\lambda ),\mathrm{C}(2,3,-1)$ and $\mathrm{D}(4,5,0)$, $|\lambda |\le 5$ are the vertices of a quadrilateral ABCD . If its area is 18 square units, then $5-6\lambda$ is equal to ______.

Let $\vec{v}=\alpha \hat{i}+2\hat{j}-3\hat{k}$, $\vec{w}=2\alpha \hat{i}+\hat{j}-\hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha >0$. If the minimum value of the scalar triple product $[\vec{u}\vec{v}\vec{w}]$ is $-\alpha \sqrt{3401}$, and …

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b}\cdot \vec{c}=0$ and $\vec{a}\times (\vec{b}\times \vec{c})=\frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b}\cdot \vec{d}=\vec{a}\cdot \vec{b}$, then …

The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the y -axis in its way and the resulting vector is $\overrightarrow{\mathrm{b}}$. Then the projection of $3\vec{a}+\sqrt{2}\vec{b}$ on $\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is

If $(2,3,9),(5,2,1),(1,\lambda ,8)$ and $(\lambda ,2,3)$ are coplanar, then the product of all possible values of $\lambda$ is:

Let $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}}$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and …

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}{|}^2=|\vec{a}{|}^2+2|\vec{b}{|}^2$, $\vec{a}\cdot \vec{b}=3$ and $|\vec{a}\times \vec{b}{|}^2=75$. Then $|\overrightarrow{\mathrm{a}}{|}^2$ is equal to ____.

Let the vectors $\vec{a}=(1+t)\hat{i}+(1-t)\hat{j}+\hat{k}$, $\overrightarrow{\mathrm{b}}=(1-\mathrm{t})\hat{\mathrm{i}}+(1+\mathrm{t})\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and …

Let a vector $\vec{a}$ has a magnitude 9. Let a vector $\vec{b}$ be such that for every $(x,y)\in R\times R-\{(0,0)\}$, the vector ( $x\vec{a}+y\vec{b}$ ) is perpendicular to the vector ( $6y\vec{a}-18x\vec{b}$ ). Then the value of $|\vec{a}\times \vec{b}|$ …

Let S be the set of all $\mathrm{a}\in \mathrm{R}$ for which the angle between the vectors $\overrightarrow{\mathrm{u}}=\mathrm{a}\left({\log }_{\mathrm{e}}\mathrm{b}\right)\hat{\mathrm{i}}-6\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ and …

Let $\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$ and $\vec{b}=3\hat{i}-5\hat{j}+4\hat{k}$ be two vectors, such that $\vec{a}\times \vec{b}=-\hat{i}+9\hat{j}+12\hat{k}$. Then the projection of $\vec{b}-2\vec{a}$ on $\vec{b}+\vec{a}$ is equal to

Let $\vec{a}=2\hat{i}-\hat{j}+5\hat{k}$ and $\vec{b}=\alpha \hat{i}+\beta \hat{j}+2\hat{k}$. If $((\vec{a}\times \vec{b})\times \hat{i})\cdot \hat{k}=\frac{23}{2}$, then $|\vec{b}\times 2\hat{j}|$ is equal to

Let ABC be a triangle such that $\overrightarrow{BC}=\vec{a}$, $\overrightarrow{CA}=\vec{b}$, $\overrightarrow{AB}=\vec{c}$, $|\vec{a}|=6\sqrt{2}$, $|\vec{b}|=2\sqrt{3}$ and $\vec{b}\cdot \vec{c}=12$.
Consider the statements:
(S1): …

A vector $\overrightarrow{a}$ is parallel to the line of intersection of the plane determined by the vectors $\hat{i},\hat{i}+\hat{j}$ and the plane determined by the vectors $\hat{i}-\hat{j},\hat{i}+\hat{k}$. The obtuse angle between $\vec{a}$ and the vector …

Let $\vec{a},\vec{b},\vec{c}$ be three non-coplanar vectors such that $\vec{a}\times \vec{b}=4\vec{c}$, $\vec{b}\times \vec{c}=9\vec{a}$ and $\vec{c}\times \vec{a}=\alpha \vec{b}$, $\alpha >0$. If $|\vec{a}|+|\vec{b}|+|\vec{c}|=36$, then $\alpha$ is equal to ____.

Let $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}$ be a vector such that $\vec{a}\times \vec{b}=2\hat{i}-\hat{k}$ and $\vec{a}\cdot \vec{b}=3$. Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :-

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\alpha \hat{k},\alpha >0$. If the projection of $\vec{a}\times \vec{b}$ on the vector $-\hat{i}+2\hat{j}-2\hat{k}$ is 30, then $\alpha$ is equal to

Let $\vec{a}=\alpha \hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=-2\hat{i}+\alpha \hat{j}+\hat{k}$, where $\alpha \in \mathbf{R}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $\sqrt{15\left({\alpha }^2+4\right)}$, the …

Let $\overrightarrow{\mathrm{a}}$ be a vector which is perpendicular to the vector $3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}$. If $\vec{a}\times (2\hat{i}+\hat{k})=2\hat{i}-13\hat{j}-4\hat{k}$, then the projection of the vector $\overrightarrow{\mathrm{a}}$ on the vector …

Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonal of a parallelogram having area $2\sqrt{2}$. Let the angle between $\vec{a}$ and $\vec{b}$ be acute. $|\vec{a}|=1$ and $|\vec{a}\cdot \vec{b}|=|\vec{a}\times \vec{b}|$. If …

If $\vec{a}=2\hat{i}+\hat{j}+3\hat{k},\vec{b}=3\hat{i}+3\hat{j}+\hat{k}$ and $\overrightarrow{\mathrm{c}}={\mathrm{c}}_1\hat{\mathrm{i}}+{\mathrm{c}}_2\hat{\mathrm{j}}+{\mathrm{c}}_3\hat{\mathrm{k}}$ are coplanar vectors and …

Let $\mathrm{A},\mathrm{B},\mathrm{C}$ be three points whose position vectors respectively are:
$\vec{a}=\hat{i}+4\hat{j}+3\hat{k}$
$\vec{b}=2\hat{i}+\alpha \hat{j}+4\hat{k},\alpha \in \mathbb{R}$
$\vec{c}=3\hat{i}-2\hat{j}+5\hat{k}$
If $\alpha$ is the smallest positive …

Let $\vec{a}=\hat{i}+\hat{j}+2\hat{k},\vec{b}=2\hat{i}-3\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{2}{\sqrt{3}}$. If …