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MathematicsSingle correct+4 / -1Question 14 of 75

JEE Main 2026 · 2 Apr · Shift 2, Mathematics question 14

Let the vectors a⃗=−i^+j^+3k^\vec a=-\hat i+\hat j+3\hat k and b⃗=i^+3j^+k^\vec b=\hat i+3\hat j+\hat k. For some λ,μ∈R\lambda,\mu\in\mathbb R, let c⃗=λa⃗+μb⃗\vec c=\lambda\vec a+\mu\vec b. If c⃗⋅(3i^−6j^+2k^)=10\vec c\cdot\left(3\hat i-6\hat j+2\hat k\right)=10 and c⃗⋅(i^+j^+k^)=−2\vec c\cdot\left(\hat i+\hat j+\hat k\right)=-2, then ∣c⃗∣2|\vec c|^2 is equal to :

Choose one option
Answer and worked solution

Official answer: B

Dot products. a⃗⋅(3,−6,2)=−3\vec a\cdot(3,-6,2)=-3, b⃗⋅(3,−6,2)=−13\vec b\cdot(3,-6,2)=-13, a⃗⋅(1,1,1)=3\vec a\cdot(1,1,1)=3, b⃗⋅(1,1,1)=5\vec b\cdot(1,1,1)=5.

Equations. −3λ−13μ=10-3\lambda-13\mu=10 and 3λ+5μ=−23\lambda+5\mu=-2, so μ=−1\mu=-1, λ=1\lambda=1.

Vector. c⃗=a⃗−b⃗=(−2,−2,2)\vec c=\vec a-\vec b=(-2,-2,2) and ∣c⃗∣2=12|\vec c|^2=12.

(B)

Topics:Dot productVector operations
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