What is ({a}-{b})× ({a}+{b})equal to?

Correct Answer is : 2(a⃗×b⃗)2({a}× {b})2(a×b)

Correct Answer is : 2(a⃗×b⃗)2({a}× {b})2(a×b)

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UPSC NDA II 2018 Mathematics Solved Paper

Question : 106 of 120

Marks: +1, -0

What is

(\vec{a}-\vec{b})\times (\vec{a}+\vec{b})

equal to?

Solution:

Concept:

The cross product is distributive over vector addition and is anti-commutative.

For any vectors

\vec{a}

and

\vec{b}

\vec{a} \times \vec{a} = \vec{0}

and

\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})

Explanation:

We expand

(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})

using the distributive property:

= \vec{a} \times \vec{a} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{b} \times \vec{b}

Since

\vec{a} \times \vec{a} = \vec{0}

and

\vec{b} \times \vec{b} = \vec{0}

, the expression simplifies to:

= \vec{0} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{0}

Using the anti-commutative property,

-\vec{b} \times \vec{a} = \vec{a} \times \vec{b}

Thus,

= \vec{a} \times \vec{b} + \vec{a} \times \vec{b} = 2(\vec{a} \times \vec{b})

Answer:

2(\vec{a} \times \vec{b})

(Option C).

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