What is {{ a}} · {{ b}} + {{ b}} · {{ c}} + {{ c}} · {{ a}} equal to?

Correct Answer is : −166-166−166

Correct Answer is : −166-166−166

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UPSC NDA 2 2016 Mathematics Solved Paper

Question Numbers: 67-68

Direction: Consider the following for the next two (02) items that follow:

Let

{\rm{\vec a}},{\rm{\;\vec b}}

and

{\rm{\vec c}}

be three vectors such that

{\rm{\vec a}} + {\rm{\vec b}} + {\rm{\vec c}} = \vec 0

, and

\left | {{\rm{\vec a}}} \right | = 10,{\rm{\;}}\left | {{\rm{\vec b}}} \right | = 6

and

\left | {{\rm{\vec c}}} \right | = 14

Question : 67 of 120

Marks: +1, -0

What is

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

equal to?

Solution:

Concept:

Use the dot product expansion of the square of the zero vector sum.

Explanation:

Given

\vec a + \vec b + \vec c = \vec 0

, take the dot product with itself.

(\vec a + \vec b + \vec c) \cdot (\vec a + \vec b + \vec c) = \vec 0 \cdot \vec 0 = 0

Expand:

|\vec a|^2 + |\vec b|^2 + |\vec c|^2 + 2(\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a) = 0

Substitute the given magnitudes:

10^2 = 100

6^2 = 36

14^2 = 196

Sum of squares:

100 + 36 + 196 = 332

Thus

332 + 2S = 0

, where

S = \vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a

2S = -332

and

S = -166

Answer:

-166

(Option B)

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