Vector Algebra in past papers.

CBSE Board Examination 2026

• Board 2026 Q13Value of $p$ for two perpendicular vectors
  Scalar product: perpendicular vectors1 mark
• Board 2026 Q14Value of $m$ for three collinear points
  Collinear points using vectors1 mark
• Board 2026 Q15$|\vec a\cdot\vec b|$ from $|\vec a|$⁠, $|\vec b|$ and $|\vec a\times\vec b|$
  Scalar and vector products1 mark
• Board 2026 Q23Vector of magnitude 14 in the direction of $\overrightarrow{QP}$
  Vector of given magnitude in a given direction2 marks
• Board 2026 Q24Diagonals of the parallelogram on $\vec a = 3\hat i - 2\hat j + 2\hat k$ and $\vec b = \hat i + 2\hat k$
  Diagonals of a parallelogram from its sides2 marks

CBSE Board Examination 2025

• Board 2025 Q2Which is true for $\vec a = 3\hat i + 2\hat j - \hat k$ and $\vec b = \hat i - \hat j + \hat k$⁠?
  Scalar product: perpendicular vectors1 mark
• Board 2025 Q11Angle between $\vec b$ and $\vec c$ when $\vec a + \vec b + \vec c = \vec 0$
  Angle between vectors from a + b + c = 01 mark
• Board 2025 Q23Area of the parallelogram with diagonals $2\hat i - \hat j + \hat k$ and $\hat i + 3\hat j - \hat k$
  Area of a parallelogram from its diagonals2 marks
• Board 2025 Q25Angle between kite strings $3\hat i + \hat j + 2\hat k$ and $2\hat i - 2\hat j + 4\hat k$⁠, or a vector of magnitude 21 opposite to $\overrightarrow{AB}$
  Angle between vectors; vector of given magnitude2 marks

CBSE Board Examination 2024

• Board 2024 Q7$\hat a \cdot \hat b$ when $\sin\theta = \frac{3}{5}$
  Scalar product of unit vectors1 mark
• Board 2024 Q15Vector from $B(3, -4, 7)$ to $A(2, -3, 5)$
  Vector joining two points1 mark
• Board 2024 Q24If $\vec a + \vec b \perp \vec a$ and $2\vec a + \vec b \perp \vec b$⁠, prove $|\vec b| = \sqrt2\,|\vec a|$
  Dot product: perpendicular vectors2 marks
• Board 2024 Q25Find $\vec{AD}$ and the area of parallelogram ABCD from $\vec{AB}$ and $\vec{DB}$
  Cross product: area of a parallelogram2 marks

CBSE Board Examination 2023

• Board 2023 Q8p for which $2\hat i + p\hat j + \hat k$ and $-4\hat i - 6\hat j + 26\hat k$ are perpendicular
  Perpendicular vectors: scalar product1 mark
• Board 2023 Q9$(\hat i \times \hat j) \cdot \hat j + (\hat j \times \hat i) \cdot \hat k$
  Products of the unit vectors i, j, k1 mark
• Board 2023 Q10$|\vec b|$ when $\vec a + \vec b = \hat i$ and $\vec a = 2\hat i - 2\hat j + 2\hat k$
  Magnitude of a vector1 mark
• Board 2023 Q22Vectors of magnitude $3\sqrt3$ collinear to $\hat i + \hat j + \hat k$
  Vectors of given magnitude collinear to a vector2 marks
• Board 2023 Q23$\vec c$ from $\overrightarrow{AC} = \frac54\overrightarrow{AB}$⁠, or are two lines perpendicular?
  Position vectors on a line; perpendicular lines2 marks

CBSE Board Examination 2022 (Term 2)

• Board 2022 T2 Q2Diagonals of parallelogram PQRS with $\overrightarrow{PQ} = 3\hat i - 2\hat j + 2\hat k$⁠, $\overrightarrow{PS} = -\hat i - 2\hat k$
  Diagonals of a parallelogram from its sides2 marks
• Board 2022 T2 Q7Show $(\vec a - 2\vec d) \parallel (2\vec b - \vec c)$⁠, or unit vectors along the diagonals and the area
  Cross product: parallel vectors, area3 marks

CBSE Sample Paper 2026-27

• Sample 2026-27 Q13Value of $u$ when $2\hat i - u\hat j + 3\hat k$ has magnitude 4
  Magnitude of a vector1 mark
• Sample 2026-27 Q14Angle of $\hat i + \hat j + \sqrt2\,\hat k$ with the positive z-axis
  Direction cosines of a vector1 mark
• Sample 2026-27 Q15A vector perpendicular to $\hat i + \hat j$ and $\hat i - \hat j$
  Vector perpendicular to two vectors1 mark
• Sample 2026-27 Q24Find $|\vec b|$ from $|\vec a|$⁠, $\vec a \cdot \vec b$ and $\vec b \times \vec a$
  Dot and cross product: finding a magnitude2 marks

CBSE Sample Paper 2025-26

• Sample 2025-26 Q13Projected length of a bird's flight on a metro track
  Projection of a vector on a line1 mark
• Sample 2025-26 Q15$(2\vec a \cdot \hat i)\hat i - (\vec b \cdot \hat j)\hat j + (\vec c \cdot \hat k)\hat k$
  Components of a vector by scalar products1 mark
• Sample 2025-26 Q20$\vec r \times (\vec a + \vec b) = \vec 0$ and $\vec r$ perpendicular to $\vec a \times \vec b$
  Vector product: parallel and perpendicular vectors1 mark
• Sample 2025-26 Q25Area of the parallelogram on $OA$ and the median $OP$ of $\Delta OAB$
  Vector product: area of a parallelogram2 marks

CBSE Sample Paper 2024-25

• Sample 2024-25 Q9$\alpha$ for an obtuse angle between $\vec p$ and $\vec q$
  Scalar product: obtuse angle between vectors1 mark
• Sample 2024-25 Q10$|\vec a - \vec b|$ when $|\vec a| = 3$⁠, $|\vec b| = 4$⁠, $|\vec a + \vec b| = 5$
  Magnitudes: |a + b| and |a − b|1 mark
• Sample 2024-25 Q24$\lambda$ for $\vec b + \lambda\vec c \perp \vec a$⁠, or the angles $\overrightarrow{BA}$ makes with the axes
  Perpendicular vectors; angles with the axes2 marks
• Sample 2024-25 Q25Diagonals and area of the parallelogram on $2\hat i - 4\hat j - 5\hat k$ and $2\hat i + 2\hat j + 3\hat k$
  Area of a parallelogram from its diagonals2 marks
• Sample 2024-25 Q28Angle between $\vec{l_1}$ and $\vec{l_2}$ and the projection, or the line perpendicular to two given lines
  Angle, projection; line perpendicular to two lines3 marks

CBSE Sample Paper 2023-24

• Sample 2023-24 Q8$\overrightarrow{EA} + \overrightarrow{EB} + \overrightarrow{EC} + \overrightarrow{ED}$ for a rhombus $ABCD$
  Addition of vectors: diagonals of a rhombus1 mark
• Sample 2023-24 Q12Vector component of $4\hat i + 6\hat j$ along $3\hat j + 4\hat k$
  Vector component (projection) of a along b1 mark
• Sample 2023-24 Q16$\lambda$ for which $2\hat i - \hat j + 2\hat k$ and $3\hat i + \lambda\hat j + \hat k$ are perpendicular
  Perpendicular vectors: dot product zero1 mark
• Sample 2023-24 Q37Case study: tug of war with three forces on a ring
  Vectors: magnitude, resultant and direction4 marks

CBSE Sample Paper 2022-23

• Sample 2022-23 Q3Area of triangle ABC from $\overrightarrow{AB} \times \overrightarrow{AC}$
  Area of a triangle using the cross product1 mark
• Sample 2022-23 Q8Projection of $3\hat i - \hat j - 2\hat k$ on $\hat i + 2\hat j - 3\hat k$
  Scalar projection of a vector on a vector1 mark
• Sample 2022-23 Q17$|\vec a - 2\vec b|$ from $|\vec a|$⁠, $|\vec b|$ and $\vec a \cdot \vec b$
  Magnitude using the dot product1 mark
• Sample 2022-23 Q23$\lambda$ for $\vec a + \vec b \perp \vec a - \vec b$⁠, or direction cosines from $6x - 12 = 3y + 9 = 2z - 2$
  Orthogonal vectors; direction ratios of a line2 marks
• Sample 2022-23 Q25Find $|\vec x|$ if $(\vec x - \vec a)\cdot(\vec x + \vec a) = 12$⁠, $\vec a$ a unit vector
  Dot product: magnitude from (x − a)·(x + a)2 marks

CBSE Sample Paper 2021-22 (Term 2)

• Sample 2021-22 T2 Q3Prove $|\hat a + \hat b| = 2\cos\frac{\theta}{2}$ for unit vectors $\hat a$⁠, $\hat b$
  Dot product: magnitude of a sum of unit vectors2 marks
• Sample 2021-22 T2 Q9If $\vec a\cdot\vec b = \vec a\cdot\vec c$ and $\vec a \times \vec b = \vec a \times \vec c$ with $\vec a \ne \vec 0$⁠, show $\vec b = \vec c$
  Dot and cross products: cancellation3 marks

CBSE Sample Paper 2020-21

• Sample 2020-21 Q10Unit vector opposite to $-\frac{3}{4}\hat j$
  Unit vector in a given direction1 mark
• Sample 2020-21 Q11Area of the triangle with sides $2\hat i$ and $-3\hat j$
  Area of a triangle by the cross product1 mark
• Sample 2020-21 Q12Angle between unit vectors $\hat a$⁠, $\hat b$ when $|\hat a + \hat b| = 1$
  Angle between two vectors (dot product)1 mark
• Sample 2020-21 Q26Parallelogram with side $\hat i - \hat j + \hat k$ and diagonal $4\hat i + 5\hat k$⁠: its area
  Area of a parallelogram by the cross product2 marks

Questions that also use Vector Algebra

Their main topic is in another chapter.

• Board 2026 Q20Perpendicular lines $x = py + q$⁠, $z = ry + s$
  Perpendicular lines in space1 mark
• Board 2025 Q29Skew lines and their shortest distance, or the ratio in which the wicketkeeper divides bowler to fielder
  Skew lines and shortest distance; section formula3 marks
• Board 2023 Q34Line through $(1, 2, -4)$ parallel to AB and the distance between them, or points on AB at 14 units from B
  Line equations; distance between parallel lines5 marks
• Sample 2026-27 Q23Parallel lines through given points: find $\frac{q}{p}$⁠; or the sides of a rectangle from its diagonals
  Parallel lines; sides of a rectangle as vectors2 marks
• Sample 2026-27 Q36Case study: laser beams in a mall, parallel and skew lines
  Lines in space: parallel lines, shortest distance4 marks
• Sample 2025-26 Q14Distance of $3\hat i + 4\hat j + 5\hat k$ from the y-axis
  Distance of a point from a coordinate axis1 mark
• Sample 2025-26 Q29Distance from $(2, -1, 3)$ to a line parallel to the z-axis, or a point of intersection and its distance from the z-axis
  Lines in space: intersection and distance3 marks
• Sample 2025-26 Q35Point of intersection of two lines and the line perpendicular to both
  Intersecting lines; a line perpendicular to both5 marks
• Sample 2023-24 Q35Image of $(1, 6, 3)$ in a line, or where two aeroplanes are closest
  Image of a point in a line; shortest distance5 marks
• Sample 2022-23 Q34Two insects on skew lines, or the path of a rocket
  Shortest distance of skew lines; point to a line5 marks
• Sample 2021-22 T2 Q10Shortest distance between two parallel lines, or the plane through $\hat i + 2\hat j - \hat k$ parallel to two lines
  Shortest distance between parallel lines3 marks
• Sample 2020-21 Q27Plane through $(1, 0, 0)$ containing the line $\vec r = \lambda\hat j$
  Vector equation of a plane containing a line2 marks
• Sample 2020-21 Q37Shortest distance between two lines and their point of intersection, or the foot of a perpendicular to a plane
  Shortest distance; point of intersection of lines5 marks