JEE (Main) · Mathematics, Class 12
Vector Algebra: JEE Main PYQ, 2020–2025
Vector Algebra (Mathematics, Class 12) has been asked 155 times in the 111 JEE (Main) shift papers from 2020–2025: about 1.4 questions a paper, 5% of all Mathematics questions. It is the 9th most-asked of 19 Mathematics chapters. The 2025 shifts had 18 from it in all. Last checked 28 October 2026.
Questions per year, all shifts
Year by year.
| Year | Questions |
|---|---|
| 2020 | 13in 14 papers |
| 2021 | 34in 26 papers |
| 2022 | 32in 24 papers |
| 2023 | 24in 12 papers |
| 2024 | 34in 18 papers |
| 2025 | 18in 17 papers |
JEE (Main) is sat in many shifts a year, each with its own paper; the counts add up every shift. 2021–2024 papers had 30 questions a subject, the others 25.
Inside the chapter
Topics asked most.
- Scalar and vector products139
- Components in two and three dimensions9
- Addition of vectors7
Try it
10 Vector Algebra questions to practise.
Our own questions, written in the JEE Main pattern and checked, each with an explanation. Pick an option to see the answer.
Q1 · Mathematics · Vector Algebra
In a pentagon ABCDE, the vector sum AB + BC + CD + DE (each a vector) is equal to which of the following?
Answer: (A)
Apply the triangle law repeatedly: AB + BC = AC, then AC + CD = AD, then AD + DE = AE. Each time the head of one vector is the tail of the next, so the sum runs from the first tail A to the last head E. It is not zero because the path A→B→C→D→E is not closed, and it is AE, not EA, because the direction is A to E.
Q2 · Mathematics · Vector Algebra
Find the vector of magnitude 39 that is directed opposite to the vector a = 3i − 4j + 12k.
Answer: (A)
A vector of magnitude m in the direction of a is m·a/|a|. Here |a| = √(9+16+144) = 13. For the opposite direction use −39/13 = −3, so the vector is −3a = −9i + 12j − 36k.
Q3 · Mathematics · Vector Algebra
Let a = 2î − ĵ + 2k̂ and b = 2î + 3ĵ + 6k̂. Which of the following vectors is along the internal bisector of the angle between a and b?
Answer: (A)
Adding unit vectors gives a rhombus whose diagonal bisects the angle. |a| = √(4+1+4) = 3 and |b| = √(4+9+36) = 7. So the bisector is along a/3 + b/7 = (7a + 3b)/21 = (20î + 2ĵ + 32k̂)/21, which is parallel to 10î + ĵ + 16k̂.
Q4 · Mathematics · Vector Algebra
If a vector a has magnitude 5 and λ = -2, what is the magnitude of the vector λa?
Answer: (A)
For a scalar λ and a vector a, |λa| = |λ| |a|. Here |λ| = |-2| = 2 and |a| = 5, so |λa| = 10. The negative sign of λ only reverses the direction of the vector and does not make the magnitude negative.
Q5 · Mathematics · Vector Algebra
What is the angle between the vectors a = i + j and b = j + k?
Answer: (B)
a · b = 0 + 1 + 0 = 1 and |a| = |b| = √2, so cos θ = 1/2 and θ = 60°.
Q6 · Mathematics · Vector Algebra
Vectors a, b, c have magnitudes 2, 3 and 6 respectively, and their directions can be chosen freely. How many integer values can |a + b + c| take?
Answer: (A)
By the triangle inequality the maximum of |a+b+c| is 2 + 3 + 6 = 11, when all three point the same way. For the minimum, |a+b| ranges over [1, 5], and c has magnitude 6, so |a+b+c| ≥ 6 − 5 = 1, attained when a and b point along −c with total magnitude 5. The magnitude varies continuously, so every value in [1, 11] occurs. The integers 1, 2, …, 11 give 11 values.
Q7 · Mathematics · Vector Algebra
Let a = 2î − ĵ + 3k̂ and b = −î + 4ĵ + k̂. Which vector c satisfies a + b + c = 0?
Answer: (C)
Add components: a + b = (2 − 1)î + (−1 + 4)ĵ + (3 + 1)k̂ = î + 3ĵ + 4k̂. For a + b + c = 0 we need c = −(a + b) = −î − 3ĵ − 4k̂.
Q8 · Mathematics · Vector Algebra
Let a and b be non-zero vectors such that a + 3b is perpendicular to 7a − 5b, and a − 4b is perpendicular to 7a − 2b. The angle between a and b is
Answer: (A)
Perpendicular vectors have zero dot product. From (a + 3b)·(7a − 5b) = 0 we get 7|a|² + 16 a·b − 15|b|² = 0. From (a − 4b)·(7a − 2b) = 0 we get 7|a|² − 30 a·b + 8|b|² = 0. Subtracting the second from the first gives 46 a·b − 23|b|² = 0, so a·b = |b|²/2. Substituting into the first equation gives 7|a|² + 8|b|² − 15|b|² = 0, so |a| = |b|. Then cos θ = a·b/(|a||b|) = (|b|²/2)/|b|² = 1/2, so θ = 60°.
Q9 · Mathematics · Vector Algebra
A vector r of magnitude 8 makes an angle of 60° with the positive z-axis. Its projection on the xy-plane makes an angle of 30° with the positive x-axis, lying in the first quadrant of that plane. What is the direction cosine of r with respect to the y-axis?
Answer: (A)
The z-component is 8 cos 60° = 4. The projection on the xy-plane has magnitude 8 sin 60° = 4√3. Splitting this projection by the 30° angle gives x = 4√3 cos 30° = 6 and y = 4√3 sin 30° = 2√3. The direction cosine with the y-axis is y/|r| = 2√3/8 = √3/4. As a check, 9/16 + 3/16 + 4/16 = 1.
Q10 · Mathematics · Vector Algebra
The vector (1/2)i + (1/2)j + ck is a unit vector, where c is a negative real number. What is the value of c?
Answer: (A)
A unit vector has magnitude 1, so (1/2)² + (1/2)² + c² = 1. Then c² = 1 − 1/2 = 1/2, so c = ±1/√2. Since c is negative, c = −1/√2.
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All 155 real JEE Main questions from Vector Algebra.
The questions from the actual papers, with the official answer and an explanation for each — chapter by chapter, or as full timed papers.
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About Vector Algebra in JEE Main.
How many questions come from Vector Algebra in JEE Main?
Vector Algebra (Mathematics, Class 12) has been asked 155 times in the 111 JEE (Main) shift papers from 2020–2025: about 1.4 questions a paper, 5% of all Mathematics questions. It is the 9th most-asked of 19 Mathematics chapters. The 2025 shifts had 18 from it in all.
Which topics of Vector Algebra are asked most in JEE Main?
Scalar and vector products (139), Components in two and three dimensions (9), Addition of vectors (7), counted over the papers from 2020–2025.
Is Vector Algebra important for JEE Main 2027?
It is in the syllabus in force, so it can be asked. By past papers it ranks 9 of 19 in Mathematics; use that to decide how much time it gets, not whether to study it.
Where can I solve Vector Algebra JEE Main previous year questions?
On Nikaalle, free: sign up and practise every past JEE Main question from Vector Algebra with an explanation for each, or sit the full papers timed. The ten questions on this page are our own, written in the same pattern.
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