Vectors

24 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q5·MCQ·Moderate
Open
Vector cross product equations
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 $\vec a\times\vec c = \vec b$ . If $\vec c\cdot(\hat i + \hat j + \hat k) = 4$ , then $\lvert\vec a + \vec c\rvert^2$ is equal to:
1. (1)33
2. (2)30
3. (3)35
4. (4)27
MathsJEE Main 21 Jan 2026, Shift 1·Q7·MCQ·Tough
Open
Cross product and collinear vectors
Let $\vec c$ and $\vec d$ be vectors such that $\lvert\vec c + \vec d\rvert = \sqrt{29}$ and $\vec c\times(2\hat i + 3\hat j + 4\hat k) = (2\hat i + 3\hat j + 4\hat k)\times\vec d$ . If $\lambda_1, \lambda_2\ (\lambda_1 > \lambda_2)$ are the possible values of $(\vec c + \vec d)\cdot(-7\hat i + 2\hat j + 3\hat k)$ , then the equation $K^2x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \dfrac{\lambda_2}{2}\right)y^2 - 8x + 12y + \lambda_2 = 0$ represents a circle, for $K$ equal to:
1. (1)4
2. (2)1
3. (3) $-1$
4. (4)2
We corrected a printing error in the original paper
Source printed “represents a circle, for $k$ equal to” (lower-case $k$ ; the equation uses $K$ ).
MathsJEE Main 21 Jan 2026, Shift 2·Q15·MCQ·Moderate
Open
Cross product magnitude
For a triangle $ABC$ , let $\vec p = \overrightarrow{BC}$ , $\vec q = \overrightarrow{CA}$ and $\vec r = \overrightarrow{BA}$ . If $\lvert\vec p\rvert = 2\sqrt3$ , $\lvert\vec q\rvert = 2$ and $\cos\theta = \dfrac{1}{\sqrt3}$ , where $\theta$ is the angle between $\vec p$ and $\vec q$ , then $\left\lvert\vec p \times \left(\vec q - 3\vec r\right)\right\rvert^2 + 3\lvert\vec r\rvert^2$ is equal to:
1. (1)340
2. (2)220
3. (3)410
4. (4)200
We corrected a printing error in the original paper
Source printed “the angle between $\vec P$ and $\vec q$ ” (capital $P$ ; the vector is $\vec p$ ).
MathsJEE Main 22 Jan 2026, Shift 1·Q1·MCQ·Moderate
Open
Projection of a vector
Let $\overrightarrow{AB} = 2\hat i + 4\hat j - 5\hat k$ and $\overrightarrow{AD} = \hat i + 2\hat j + \lambda\hat k$ , $\lambda \in \mathbb{R}$ . Let the projection of the vector $\vec v = \hat i + \hat j + \hat k$ on the diagonal $\overrightarrow{AC}$ of the parallelogram $ABCD$ be of length one unit. If $\alpha, \beta$ , where $\alpha > \beta$ , be the roots of the equation $\lambda^2x^2 - 6\lambda x + 5 = 0$ , then $2\alpha - \beta$ is equal to
1. (1)1
2. (2)4
3. (3)3
4. (4)6
We corrected a printing error in the original paper
Source printed “ $2\hat i + 4\hat j - 5k$ ” and “ $\hat i + 2\hat j + \lambda k$ ” (unit vector $\hat k$ printed without its hat).
MathsJEE Main 22 Jan 2026, Shift 2·Q8·MCQ·Moderate
Open
Cross product and dot product
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 $\lvert\vec c\rvert = \sqrt{53}$ , then the maximum possible value of $\left(\vec c\cdot\vec d\right)^2$ is equal to:
1. (1)26
2. (2)104
3. (3)208
4. (4)52
MathsJEE Main 22 Jan 2026, Shift 2·Q21·Numerical·Moderate
Open
Angle between vectors
Let a vector $\vec a = \sqrt2\hat i - \hat j + \lambda\hat k$ , $\lambda > 0$ , make an obtuse angle with the vector $\vec b = -\lambda^2\hat i + 4\sqrt2\hat j + 4\sqrt2\hat k$ and an angle $\theta$ , $\dfrac{\pi}{6} < \theta < \dfrac{\pi}{2}$ , with the positive $z$ -axis. If the set of all possible values of $\lambda$ is $(\alpha, \beta) - \{\gamma\}$ , then $\alpha + \beta + \gamma$ is equal to .
MathsJEE Main 23 Jan 2026, Shift 1·Q3·MCQ·Easy–moderate
Open
Vector triple product
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 $\left(\vec a - \vec b\right)\cdot\vec d$ is equal to:
1. (1)4
2. (2) $-4$
3. (3) $-2$
4. (4)2
MathsJEE Main 23 Jan 2026, Shift 2·Q3·MCQ·Easy–moderate
Open
Cross product and projection
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 $9p^2$ is equal to:
1. (1)9
2. (2)6
3. (3)4
4. (4)12
MathsJEE Main 23 Jan 2026, Shift 2·Q10·MCQ·Moderate
Open
Vector cross product equations
Let $\vec a, \vec b, \vec c$ be three vectors such that $\vec a\times\vec b = 2(\vec a\times\vec c)$ . If $\lvert\vec a\rvert = 1$ , $\lvert\vec b\rvert = 4$ , $\lvert\vec c\rvert = 2$ , and the angle between $\vec b$ and $\vec c$ is $60^\circ$ , then $\lvert\vec a\cdot\vec c\rvert$ is:
1. (1)2
2. (2)4
3. (3)0
4. (4)1
MathsJEE Main 24 Jan 2026, Shift 1·Q5·MCQ·Easy–moderate
Open
Cross product and dot product
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 $\vec d$ be a vector such that $\lvert\vec d - \vec a\rvert = \sqrt{11}$ , $\lvert\vec c \times \vec d\rvert = 3$ and the angle between $\vec c$ and $\vec d$ is $\dfrac{\pi}{4}$ . Then $\vec a \cdot \vec d$ is equal to
1. (1)11
2. (2)3
3. (3)0
4. (4)1
MathsJEE Main 24 Jan 2026, Shift 2·Q4·MCQ·Moderate
Open
Projection of a vector
Let $\vec a = 2\hat i - \hat j - \hat k$ , $\vec b = \hat i + 3\hat j - \hat k$ and $\vec c = 2\hat i + \hat j + 3\hat k$ . Let $\vec v = \vec a + \lambda\vec b$ , $\lambda \in \mathbb{R}$ , be the vector in the plane of the vectors $\vec a$ and $\vec b$ , such that the length of its projection on the vector $\vec c$ is $\dfrac{1}{\sqrt{14}}$ . Then $\lvert\vec v\rvert$ is equal to
1. (1) $\dfrac{\sqrt{21}}{2}$
2. (2)13
3. (3) $\dfrac{\sqrt{35}}{2}$
4. (4)7
We corrected a printing error in the original paper
Assumption “ $\vec v = \vec a + \lambda\vec b$ , $\lambda \in \mathbb{R}$ ” added; the source states only that $\vec v$ lies in the plane of $\vec a$ and $\vec b$ , which leaves $\lvert\vec v\rvert$ undetermined (the source prints “Ans. (Bonus)” and “NTA (3)”); key (3) holds when the coefficient of $\vec a$ is 1.
MathsJEE Main 24 Jan 2026, Shift 2·Q19·MCQ·Moderate
Open
Vector cross product equations
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 $2\left(\vec a\times\vec c\right) + 3\left(\vec b\times\vec c\right) = \vec 0 \quad\text{and}\quad \left(\vec a - \vec b\right)\cdot\vec c = -97,$ then $\left\lvert\vec c\times\hat k\right\rvert^2$ is equal to
1. (1)193
2. (2)233
3. (3)218
4. (4)205
We corrected a printing error in the original paper
Source printed “ $2\hat i - 5\hat j + 5k$ ”, “ $\hat i - \hat j + 3k$ ” and “ $\lvert\vec c\times k\rvert^2$ ” (unit vector $\hat k$ printed without its hat).
MathsJEE Main 28 Jan 2026, Shift 1·Q19·MCQ·Easy–moderate
Open
Magnitude and dot product
For three unit vectors $\vec a, \vec b, \vec c$ satisfying $\lvert\vec a - \vec b\rvert^2 + \lvert\vec b - \vec c\rvert^2 + \lvert\vec c - \vec a\rvert^2 = 9$ and $\lvert 2\vec a + k\vec b + k\vec c\rvert = 3$ , the positive value of $k$ is:
1. (1)3
2. (2)6
3. (3)4
4. (4)5
MathsJEE Main 28 Jan 2026, Shift 1·Q25·Numerical·Tough
Open
Angle bisector in a triangle
Let $PQR$ be a triangle such that $\overrightarrow{PQ} = -2\hat i - \hat j + 2\hat k$ and $\overrightarrow{PR} = a\hat i + b\hat j - 4\hat k$ , $a, b \in \mathbb{Z}$ . Let $S$ be the point on $QR$ , which is equidistant from the lines $PQ$ and $PR$ . If $\lvert\overrightarrow{PR}\rvert = 9$ and $\overrightarrow{PS}$ is parallel to $\hat i - 7\hat j + 2\hat k$ , then the value of $3a - 4b$ is .
We corrected a printing error in the original paper
Source printed “ $\overrightarrow{PS} = \hat i - 7\hat j + 2\hat k$ ”; corrected to the intended “ $\overrightarrow{PS}$ is parallel to $\hat i - 7\hat j + 2\hat k$ ”: the printed vector has length $3\sqrt6$ and puts $S$ off the line $QR$ (no such triangle; the source marks the question Bonus), while its direction is the internal bisector for $a = 7$ , $b = -4$ , which the key 37 uses.
MathsJEE Main 28 Jan 2026, Shift 2·Q17·MCQ·Tough
Open
Angle bisector and triangle area
Let $P$ be a point in the plane of the vectors $\overrightarrow{AB} = 3\hat i + \hat j - \hat k$ and $\overrightarrow{AC} = \hat i - \hat j + 3\hat k$ such that $P$ is equidistant from the lines $AB$ and $AC$ . If $\left\lvert\overrightarrow{AP}\right\rvert = \dfrac{\sqrt5}{2}$ , then the area of the triangle $ABP$ is :
1. (1)2
2. (2) $\dfrac32$
3. (3) $\dfrac{\sqrt{30}}{4}$
4. (4) $\dfrac{\sqrt{26}}{4}$
We corrected a printing error in the original paper
Source printed “in the plane of the vector” (two vectors follow).
MathsJEE Main 2 Apr 2026, Shift 1·Q14·MCQ·Moderate
Open
Maximum of magnitude expression
If $\vec a$ and $\vec b$ are two vectors such that $\lvert\vec a\rvert = 2$ and $\lvert\vec b\rvert = 3$ , then the maximum value of $3\left\lvert\left(3\vec a + 2\vec b\right)\right\rvert + 4\left\lvert\left(3\vec a - 2\vec b\right)\right\rvert$ is:
1. (1)30
2. (2)36
3. (3)60
4. (4)72
MathsJEE Main 2 Apr 2026, Shift 2·Q14·MCQ·Easy–moderate
Open
Dot product and linear combination
Let the vectors $\vec a = -\hat i + \hat j + 3\hat k$ and $\vec b = \hat i + 3\hat j + \hat k$ . For some $\lambda, \mu \in \mathbb{R}$ , let $\vec c = \lambda\vec a + \mu\vec b$ . If $\vec c\cdot(3\hat i - 6\hat j + 2\hat k) = 10$ and $\vec c\cdot(\hat i + \hat j + \hat k) = -2$ , then $\lvert\vec c\rvert^2$ is equal to:
1. (1)8
2. (2)12
3. (3)14
4. (4)15
MathsJEE Main 2 Apr 2026, Shift 2·Q16·MCQ·Moderate
Open
Angle between two vectors
Two adjacent sides of a parallelogram $PQRS$ are given by $\overrightarrow{PQ} = \hat j + \hat k$ and $\overrightarrow{PS} = \hat i - \hat j$ . If the side $PS$ is rotated about the point $P$ by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side $PQ$ , then $\sin^2\left(\dfrac{5\alpha}{2}\right) - \sin^2\left(\dfrac{\alpha}{2}\right)$ is equal to:
1. (1) $\dfrac12$
2. (2) $\dfrac{\sqrt3}{2}$
3. (3) $\dfrac{\sqrt3}{4}$
4. (4) $\dfrac{2\sqrt3}{5}$
MathsJEE Main 4 Apr 2026, Shift 2·Q15·MCQ·Moderate
Open
Scalar and vector products
Let $\hat u$ and $\hat v$ be unit vectors inclined at an acute angle such that $\lvert\hat u \times \hat v\rvert = \dfrac{\sqrt3}{2}$ . If $\vec A = \lambda\hat u + \hat v + (\hat u \times \hat v)$ . Then $\lambda$ is equal to:
1. (1) $\dfrac43\left(\vec A\cdot\hat u\right) - \dfrac23\left(\vec A\cdot\hat v\right)$
2. (2) $\dfrac23\left(\vec A\cdot\hat u\right) - \dfrac13\left(\vec A\cdot\hat v\right)$
3. (3) $\dfrac43\left(\vec A\cdot\hat u\right) + \dfrac23\left(\vec A\cdot\hat v\right)$
4. (4) $\left(\vec A\cdot\hat u\right) - \dfrac12\left(\vec A\cdot\hat v\right)$
MathsJEE Main 5 Apr 2026, Shift 1·Q14·MCQ·Moderate
Open
Vector cross product equations
Let $\vec a = \sqrt7\,\hat i + \hat j - \hat k$ and $\vec b = \hat j + 2\hat k$ . If $\vec r$ is a vector such that $\vec r\times\vec a + \vec a\times\vec b = \vec 0$ and $\vec r\cdot\vec a = 0$ , then $\lvert 3\vec r\rvert^2$ is equal to:
1. (1)44
2. (2)54
3. (3)86
4. (4)132
MathsJEE Main 5 Apr 2026, Shift 2·Q16·MCQ·Easy–moderate
Open
Position vectors of points
Let $O$ be the origin, $\overrightarrow{OP} = \vec a$ and $\overrightarrow{OQ} = \vec b$ . If $R$ is the point on $\overrightarrow{OP}$ such that $\overrightarrow{OP} = 5\overrightarrow{OR}$ , and $M$ is the point such that $\overrightarrow{OQ} = 5\overrightarrow{RM}$ , then $\overrightarrow{PM}$ is equal to:
1. (1) $\dfrac15\left(\vec a - 4\vec b\right)$
2. (2) $\dfrac15\left(\vec b - 4\vec a\right)$
3. (3) $\dfrac15\left(-\vec a + 4\vec b\right)$
4. (4) $\dfrac15\left(-\vec b + 4\vec a\right)$
MathsJEE Main 6 Apr 2026, Shift 1·Q23·Numerical·Moderate
Open
Vector triple product
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 .
MathsJEE Main 6 Apr 2026, Shift 2·Q15·MCQ·Easy–moderate
Open
Area of triangle by cross product
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 $\left(2\vec a + 3\vec b\right)$ and $\left(\vec a - \vec b\right)$ is:
1. (1)450
2. (2)900
3. (3)1800
4. (4)2400
MathsJEE Main 8 Apr 2026, Shift 2·Q14·MCQ·Moderate
Open
Vector cross product equations
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 $2(\vec a\times\vec b) + 3(\vec b\times\vec c) = \vec 0$ . If $\vec a\cdot\vec c = 15$ , then $\vec c\cdot(\hat i + \hat j - 3\hat k)$ is equal to:
1. (1) $-6$
2. (2) $-5$
3. (3) $-4$
4. (4) $-3$