If | {a}+{b} | = | {a}-{b} | , then which one of the following is correct?

Correct Answer is : a⃗{a}a is perpendicular to b⃗{b}b

Correct Answer is : a⃗{a}a is perpendicular to b⃗{b}b

Test Index

UPSC NDA I 2015 Mathematics Paper

Question : 105 of 120

Marks: +1, -0

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

, then which one of the following is correct?

Solution:

Concept:

The dot product

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta

is zero when

\theta = 90^\circ

, meaning

\vec{a}

is perpendicular to

\vec{b}

Explanation:

We are given

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

Square both sides:

|\vec{a}+\vec{b}|^2 = |\vec{a}-\vec{b}|^2

Use

|\vec{v}|^2 = \vec{v} \cdot \vec{v}

to expand:

(\vec{a}+\vec{b}) \cdot (\vec{a}+\vec{b}) = (\vec{a}-\vec{b}) \cdot (\vec{a}-\vec{b})

This gives

\vec{a}\cdot\vec{a} + \vec{b}\cdot\vec{b} + 2\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{a} + \vec{b}\cdot\vec{b} - 2\vec{a}\cdot\vec{b}

Cancel

\vec{a}\cdot\vec{a} + \vec{b}\cdot\vec{b}

from both sides:

2\vec{a}\cdot\vec{b} = -2\vec{a}\cdot\vec{b}

Thus

4\vec{a}\cdot\vec{b} = 0

, so

\vec{a}\cdot\vec{b} = 0

A zero dot product implies the vectors are perpendicular (angle

90^\circ

Answer:

\vec{a}

is perpendicular to

\vec{b}

. Hence, option C is correct.

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