Let the vectors vec a = -hat i +hat j +3hat k and vec b = hat i +3hat j +hat k . For some λ ,μ in mathbb R let vec c = λ vec a +μ vec b . If vec c . (3hat i - 6hat j +2hat k ) = 10 and vec c . (hat i +hat j +hat k ) = -2 , then vec c 2 is equal to:

Mathematics · Vector Algebra

JEE Main 2026 — 2 April, Evening Shift — Question 36

Let the vectors $\vec{\mathrm{a}} = -\hat{\mathrm{i}} +\hat{\mathrm{j}} +3\hat{\mathrm{k}}$ and $\vec{\mathrm{b}} = \hat{\mathrm{i}} +3\hat{\mathrm{j}} +\hat{\mathrm{k}}$ . For some $\lambda ,\mu \in \mathbb{R}$ let $\vec{\mathrm{c}} = \lambda \vec{\mathrm{a}} +\mu \vec{\mathrm{b}}$ . If $\vec{\mathrm{c}}.\left(3\hat{\mathrm{i}} - 6\hat{\mathrm{j}} +2\hat{\mathrm{k}}\right) = 10$ and $\vec{\mathrm{c}}.\left(\hat{\mathrm{i}} +\hat{\mathrm{j}} +\hat{\mathrm{k}}\right) = -2$ , then $|\vec{\mathrm{c}} |^{2}$ is equal to:

Option A:
8
Option B:
12
Correct
Option C:
14
Option D:
15

Answer: B

Step-by-step solution

$\overrightarrow{\mathrm{c}}=\lambda \overrightarrow{\mathrm{a}}+\mu \overrightarrow{\mathrm{b}}$ $\overrightarrow{\mathrm{c}}=(\mu-\lambda) \hat{\mathrm{i}}+(\lambda+3 \mu) \hat{\mathrm{j}}+(3 \lambda+\mu) \hat{\mathrm{k}}$ $\overrightarrow{\mathrm{c}} \cdot(3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})=10$ $3(\mu-\lambda)-6(\lambda+3 \mu)+2(3 \lambda+\mu)=10$

-3 \lambda-13 \mu=10 \end{gathered}$$ $\overrightarrow{\mathrm{c}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=-2$ $$\begin{gathered} 5 \mu+3 \lambda=-2 \end{gathered}$$ Solving (1) & (2) $\lambda=1, \mu=-1$ $\overrightarrow{\mathrm{c}}=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ $|\overrightarrow{\mathrm{c}}|^{2}=12$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 2 April, Evening Shift
Subject: Mathematics
Chapter: Vector Algebra
Topic: Scalar or Dot Product of Two Vectors