Let vec a =-hat i +hat j +2 hat k , vec b =hat i -hat j -3 hat k , vec c =vec a × vec b and overrightarrow d =overrightarrow c × overrightarrow a . Then (overrightarrow a -overrightarrow b ) × overrightarrow d is equal to :

Mathematics · Vector Algebra

JEE Main 2026 — 23 January, Morning Shift — Question 3

Let $\vec{a}=-\hat{i}+\hat{j}+2 \hat{k}, \vec{b}=\hat{i}-\hat{j}-3 \hat{k}, \vec{c}=\vec{a} \times \vec{b}$ and $\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$ . Then $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{d}}$ is equal to :

Option A:
$4$
Option B:
$-4$
Option C:
$-2$
Correct
Option D:
$2$

Answer: C

Step-by-step solution

$\vec{d}=(\vec{a} \times \vec{b}) \times \vec{a}$

$\overrightarrow{\mathrm{d}}=\left(\mathrm{a}^{2}\right) \overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}) \overrightarrow{\mathrm{a}}$

$\overrightarrow{\mathrm{d}}=6 \overrightarrow{\mathrm{~b}}+8 \overrightarrow{\mathrm{a}}$

$(\vec{a}-\vec{b}) \cdot \vec{d}-=(\vec{a}-\vec{b}) \cdot(6 \vec{b}+8 \vec{a})$

$=8 \mathrm{a}^{2}-6 \mathrm{~b}^{2}-2 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}$

$=48-66+16=-2$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 23 January, Morning Shift
Subject: Mathematics
Chapter: Vector Algebra
Topic: Vector or Cross Product of Two Vectors