Let vec a =2 hat i -3 hat j +hat k , vec b =3 hat i +2 hat j +5 hat k and a vector vec c be such that (vec a -vec c ) × vec b =-18 hat i -3 hat j +12 hat k and vec a × vec c =3 . If vec b × vec c =vec d , then vec a × vec d is equal to :

Mathematics · Vector Algebra

JEE Main 2025 — 2 April, Evening Shift — Question 36

Let $\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}+5 \hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a}-\vec{c}) \times \vec{b}=-18 \hat{i}-3 \hat{j}+12 \hat{k}$

and $\vec{a} \cdot \vec{c}=3$ . If $\vec{b} \times \vec{c}=\vec{d}$ , then $|\vec{a} \cdot \vec{d}|$ is equal to :

Option A:
9
Option B:
12
Option C:
18
Option D:
15
Correct

Answer: D

Step-by-step solution

$(\vec{a}-\vec{c}) \times \vec{b}=-18 \hat{i}-3 \hat{j}+12 \hat{k}$

$\vec{a} \times \vec{b}-\vec{c} \times \vec{b}=-18 \hat{i}-3 \hat{j}+12 \hat{k}$

$\Rightarrow \vec{a} \times \vec{b}+\vec{d}=-18 \hat{i}-3 \hat{j}+12 \hat{k}$

Take dot with $\vec{a}$

$\vec{a} \cdot \vec{d}=-36+9+12=-15$

$\therefore|\vec{a} \cdot \vec{d}|=15$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 2 April, Evening Shift
Subject: Mathematics
Chapter: Vector Algebra
Topic: Applications of Vectors