JEE Main Vector Important Questions

JEE Main Vector Important Questions - Getting ready to give the JEE Mains 2026? But sceptical about how to ace the JEE Main Mathematics Vector Section? Do not worry, you can ace the JEE Main 2026 exams by practicing and solving the vector section of the JEE Mains Previous Year Question Papers. While attempting the vector part of the question paper, one needs to have a solid understanding of the principles of Vector Algebra and efficient problem-solving skills. There are many topics that are included in the Vector section of the question paper, like Vectors, Types of Vectors, Vector Algebra, Section Formula, and Product of Two Vectors. To solve these questions with ease, being consistent in practicing the problems is a must.

The National Testing Agency or NTA has released the JEE Main 2026  Exam Dates on the official website. JEE Main 2026 date session 1 is January 22 to 30, 2026, and the JEE Main session 2 exam date is April 2026. Given below is all the information and all the questions you need to successfully solve the vector algebra section of the JEE Main 2026 Mathematics Question Paper.

Also Read: JEE Main 2026 Admit Card

JEE Mains Vector Important Questions

Given below are some important questions along with their step-by-step solution for students to practice and ace their JEE Mains. These questions will also help you in managing your time when taking the JEE Main Mathematics Vector Section.

Question 1: If the unit vectors are a,b, and c, then |a − b|2 + |b − c|2 + |c − a|2 does not exceed -

A) 4

B) 9

C) 8

D) 6

Solution:

|a − b|2 + |b − c|2 + |c − a|2 = 2 (a2 + b2 + c2) − 2 (a * b + b * c + c * a)

= 2 * 3 − 2 (a * b + b * c + c * a)

= 6 − {(a + b + c)2 − a2− b2 − c2}

= 9 − |a + b + c| 2 ≤ 9

Question 2: Let b = 4i + 3j and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane having projections 1 and 2 along b and c respectively are given by _________.

Solution:

Let r = λb + μc and c = ± (xi + yj).
Since c and b are perpendicular, we have 4x + 3y = 0

⇒ c = ±x (i − 43j), {Because, y = [−4 / 3]x}

Now, the projection of r on b = [r. b] / [|b|] = 1

⇒ [(λb + μc) . b] / [|b|]

= [λb. B] / [|b|] = 1

⇒ λ = 1 / 5

Again, projection of r on c = [r. c] / [|c|] = 2

This gives μx = [6 / 5]

⇒ r = [1 / 5] (4i + 3j) + [6 / 5] (i − [4 / 3]j)

= 2i−j or

r = [1 / 5] (4i + 3j) − [6 / 5] (i − [4 / 3]j)

= [−2 / 5] i + [11 / 5] j

Question 3: A vector has components 2p and 1 with respect to a rectangular cartesian system. Through a certain angle, the system is rotated about the origin in the anti-clockwise sense. If a has components p + 1 and 1 with respect to the new system, then find p.

Solution:

If x, y are the original components; X, Y the new components, and α is the angle of rotation, then x = X cosα − Y sinα and y = X sinα + Y cosα

Therefore, 2p = (p + 1) cosα − sinα and 1 = (p + 1) sinα + cosα

Squaring and adding, we get 4p2 + 1 = (p + 1)2 + 1

⇒ p + 1 = ± 2p

⇒ p = 1 or −1 / 3

Question 4: If b and c are any two non-collinear unit vectors and a is any vector, then (a . b) b + (a . c) c + [a. (b × c) / |b × c|] (b × c) = ___________.

Solution:

Let i be a unit vector in the direction of b, and j in the direction of c.

Note that b = i and c = j

We have b × c = |b| |c| sinαk = sinαk, where k is a unit vector perpendicular to b and c. ⇒ |b × c| = sinα

⇒ k = [b × c] / [|b × c|]

Any vector a can be written as a linear combination of i, j, and k.

Let a = a1i + a2j + a3k

Now a . b = a . i = a1, a . c = a . j = a2 and {[a] . [b × c] / [|b × c|]} = a . k = a3

Thus, (a . b) b + (a . c) c + {[a] . [(b × c) / |b × c|] * [(b × c)|]}

= a1b + a2c + a3 [b × c] / [|b × c|]

= a1i + a2j + a3k

= a

Question 5: Let p, q, and r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies equation p × {(x − q) × p} + q × {(x − r) × q} + r × {(x − p) × r} = 0, then x is given by ____________.

Solution:

|p| = |q| = |r| = c, (say) and

p . q = 0 = p . r = q . r

p × |( x − q) × p |+ q × |(x − r) × q| + r × |( x − p) × r| = 0

⇒ (p . p) (x − q) − {p . (x − q)} p + . . . . . . . . . = 0

⇒ c2 (x − q + x − r + x − p) − (p . x) p − (q . x) q − (r . x) r = 0

⇒ c2 {3x − (p + q + r)} − [(p . x) p + (q . x) q + (r . x) r] = 0

which is satisfied by x = [1 / 2] (p + q+ r).

Question 6: If a vector P−→ making angles a, b, and g respectively with the X, Y, and Z axes respectively. Then sin2α+sin2β+sin2γ=

Solution : sin2α+sin2β+sinγ =1−cos2α+1−cos2β+1−cos2γ

=3−(cos2α+cos2β+cos2γ)

=3−1=2

Question 7: If the resultant of n forces of different magnitudes acting at a point is zero, then the minimum value of n is

A) 1
B) 2

C) 3

D) 4

Correct Answer: C

Solution :

If vectors are of equal magnitude then two vectors can give zero resultant, if they work in opposite directions. But if the vectors are of different magnitudes then a minimum of three vectors are required to give zero resultant.

Question 8: Let a, b, and c be vectors with magnitudes 3, 4, and 5 respectively, and a + b + c = 0, then the values of a . b + b. c + c . a is ________.

Solution:

Since a + b + c = 0

On squaring both sides, we get

|a|2 + |b|2 + |c|2 + 2 (a . b + b . c + c . a) = 0

⇒ 2 (a . b + b . c + c . a) = − (9 + 16 + 25)

⇒ a . b + b . c + c . a = −25

Question 9. A unit vector “a” makes an angle π / 4 with a z-axis. If a + i + j is a unit vector, then a is equal to _________.

Solution:

Let a = li + mj + nk, where l2 + m2 + n2 = 1. a makes an angle π / 4 with a z−axis.

Hence, n = 1 / √2, l2 + m2 = 1 / 2 …..(i)

Therefore, a = li + mj + k / √2

a + i + j = (l + 1) i + (m + 1) j + k / √2

Its magnitude is 1, hence (l + 1)2 + (m + 1)2 = 1 / 2 …..(ii)

From (i) and (ii),

2lm = 1 / 2

⇒ l = m = −1 / 2

Hence, a = [−i / 2] − [j / 2] + [k / √2].

Question 10. The magnitudes of mutually perpendicular forces a, b, and c are 2, 10, and 11, respectively. Then the magnitude of its resultant is ______.

Solution:

R = √[22 + 102 + 112].

= √[4 + 100 + 121]

= 15

Also, check

Benefits of Solving JEE Main Vector Important Questions

Solving the JEE Main Vector important questions along with their solutions will be beneficial for preparing for the JEE Main 2026 exam, and eventually successfully clearing the exam.

Given below are some advantages of solving JEE Main Vector important questions:

• By being consistent in practicing the vector important questions, one gains a thorough understanding of the topic.
• Solving these important questions will help one develop quick logical thinking skills.
• Get an idea about the JEE Main 2026 exam pattern.
• One can recall all the important formulas by practicing the JEE Main Vector important questions.
• By constant practicing, one can realize, analyze, and correct their mistakes.

JEE Main Mathematics Question Paper

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About Vector Algebra

Vector Algebra is one of the most significant topics of Algebra. There are two types of physical quantities, the first is scalar and the second quantity is vector. Through vector algebra, we get to learn about electromagnetic fields, gravitational fields, fluid flow, and various topics of engineering and Mathematics. By using vector algebra one can solve differential equations and partial differential equations.

Vector Definition -  An object that has both magnitudes and direction is defined as a vector.

Vector is usually represented by an arrow, the length of the arrow shows the magnitude. When two vectors have equal magnitudes and equal direction, then they are said to be equal.

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