Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202623 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Vector Algebra

JEE Main 2026 · 23 January · Evening Shift · Mathematics · Vector Algebra

Let a , b , c be three vectors such that a × b =2( a × c ) . If | a...

Let a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} be three vectors such that a⃗×b⃗=2(a⃗×c⃗)\vec{a} \times \vec{b}=2(\vec{a} \times \vec{c}). If ∣a⃗∣=1,∣b⃗∣=4,∣c⃗∣=2|\vec{a}|=1,|\vec{b}|=4,|\vec{c}|=2, and the angle between b⃗\vec{b} and c⃗\vec{c} is 60∘60^{\circ}, then ∣a⃗⋅c⃗∣|\vec{a} \cdot \vec{c}| is equal to

Options

  1. A1
  2. B2
  3. C0
  4. D4

Correct answer

A. 1

Step-by-step solution

a⃗×b⃗=2(a⃗×c⃗)⇒a⃗×(b⃗−2c⃗)=0⃗\vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c}) \Rightarrow \vec{a} \times (\vec{b} - 2\vec{c}) = \vec{0}, so b⃗−2c⃗=ka⃗\vec{b} - 2\vec{c} = k\vec{a}.
b⃗⋅c⃗=4⋅2⋅cos⁡60°=4\vec{b} \cdot \vec{c} = 4 \cdot 2 \cdot \cos 60° = 4.
Dot with c⃗\vec{c}: 4=2(4)+k(a⃗⋅c⃗)⇒k(a⃗⋅c⃗)=−44 = 2(4) + k(\vec{a} \cdot \vec{c}) \Rightarrow k(\vec{a} \cdot \vec{c}) = -4.
Dot with a⃗\vec{a}: a⃗⋅b⃗=2(a⃗⋅c⃗)+k\vec{a} \cdot \vec{b} = 2(\vec{a} \cdot \vec{c}) + k.
Dot with b⃗\vec{b}: 16=8+ka⃗⋅b⃗=8+k(−8k+k)=k2⇒k=±416 = 8 + k\vec{a} \cdot \vec{b} = 8 + k\left(-\frac{8}{k} + k\right) = k^2 \Rightarrow k = \pm 4.
a⃗⋅c⃗=−4k=∓1⇒∣a⃗⋅c⃗∣=1\vec{a} \cdot \vec{c} = -\frac{4}{k} = \mp 1 \Rightarrow |\vec{a} \cdot \vec{c}| = 1.
Practice Vector Algebra on Quantrex Academy →

More from Vector Algebra

Let a , b be two vectors, and let P, Q and R be the points with position vectors a , b… JEE Advanced 2026Let a = 4 i - j + 3 k , b = 10 i + 2 j - k and a vector c be such that 2( a × b ) + 3( b… JEE Main 2026Let a = 2 i + 3 j + 3 k and b = 6 i + 3 j + 3 k . Then the square of the area of the… JEE Main 2026If a = i + j + k , b = j - k and c be three vectors such that a × c = b and a · c = 3… JEE Main 2026Let O be the origin, OP = a and OQ = b . If R is the point on OP such that OP = 5 OR… JEE Main 2026Let a = 7 i + j - k and b = j + 2 k . If r is a vector such that r × a + a × b = 0 and r… JEE Main 2026Let u and v be unit vectors inclined at an acute angle such that | u × v |= 3 2 . If A =λ… JEE Main 2026Let a_k = ( θ_k) i + j and b_k = i - ( θ_k) j , where θ_k = 2^ k-1 π 2^n + 1 , for some n… JEE Main 2026Let the vectors a = - i + j + 3 k and b = i + 3 j + k . For some λ, μ ∈ ℝ , let c = λ a +… JEE Main 2026Two adjacent sides of a parallelogram PQRS are given by PQ = j + k and PS = i - j . If… JEE Main 2026If a and b are two vectors such that | a | = 2 and | b | = 3 , then the maximum value of… JEE Main 2026Let P be a point in the plane of the vectors A B =3 i + j - k and A C = i - j +3 k such… JEE Main 2026For three unit vectors a , b , c satisfying | a - b |^ 2 +| b - c |^ 2 +| c - a |^ 2 =9… JEE Main 2026Let P Q R be a triangle such that P Q =-2 i - j +2 k and PR =a i +b j -4 k , a, b ∈ ℤ .… JEE Main 2026Let a =2 i -5 j +5 k and b = i - j +3 k . If c is a vector such that 2( a × c )+3( b × c… JEE Main 2026Let a =2 i - j - k , b = i +3 j - k and c =2 i + j +3 k . Let v be the vector in the… JEE Main 2026 Full Vector Algebra list Vector Algebra JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs