If a⃗ and b⃗ are two vectors then the value of (a⃗+b⃗) ×(a⃗-b⃗) is (A) 2(b⃗ ×a⃗) (B) -2(b⃗ ×a⃗)

Name: Problem 34, Chapter 1: A Complete Resource Book in Physics for JEE Main
Uploaded: 2021-09-09T11:43:53-08:00
Duration: 2 min 57 s
Channel: Chandana Deeksha
Description: If a⃗ and b⃗ are two vectors then the value of (a⃗+b⃗) ×(a⃗-b⃗) is (A) 2(b⃗ ×a⃗) (B) -2(b⃗ ×a⃗) (C) b⃗ ×a⃗ (D) a⃗ ×b⃗

step 1 So here we are required to simplify this given vector expression, which is cross product of vect

Question

If $\vec{a}$ and $\vec{b}$ are two vectors then the value of $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})$ is (A) $2(\vec{b} \times \vec{a})$ (B) $-2(\vec{b} \times \vec{a})$ (C) $\vec{b} \times \vec{a}$ (D) $\vec{a} \times \vec{b}$

   If $\vec{a}$ and $\vec{b}$ are two vectors then the value of $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})$ is
(A) $2(\vec{b} \times \vec{a})$
(B) $-2(\vec{b} \times \vec{a})$
(C) $\vec{b} \times \vec{a}$
(D) $\vec{a} \times \vec{b}$

A Complete Resource Book in Physics for JEE Main

Sanjeev Kumar 1st Edition

Chapter 1, Problem 34

Instant Answer

Solved by Expert Chandana Deeksha

09/09/2021

Step 1

First, we can distribute the cross product using the properties of cross products: $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b}) = \vec{a} \times \vec{a} - \vec{a} \times \vec{b} + \vec{b} \times \vec{a} - \vec{b} \times \vec{b}$ Now, we know that the cross product Show more…

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If $\vec{a}$ and $\vec{b}$ are two vectors then the value of $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})$ is (A) $2(\vec{b} \times \vec{a})$ (B) $-2(\vec{b} \times \vec{a})$ (C) $\vec{b} \times \vec{a}$ (D) $\vec{a} \times \vec{b}$

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Key Concepts

Zero Property of the Cross Product

Taking the cross product of a vector with itself always results in the zero vector, as there is no vector perpendicular to a vector along its own direction. This principle is essential when simplifying cross product expressions involving identical vectors.

Anticommutative Property

The cross product is anticommutative, meaning that swapping the order of the vectors changes the sign of the result (i.e., a × b = -b × a). This fact is often used to rearrange terms in vector cross product expressions to facilitate simplification.

Distributive Property

The cross product is distributive over vector addition, which means that when you have a cross product of a sum of vectors with another vector, it can be expanded as the sum of individual cross products. This property is crucial for simplifying expressions with sums inside a cross product.

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