Test Index

MHTCET 23 Apr 2025 Shift 2 Paper

Question : 126 of 150

Marks: +1, -0

Let

\bar{a}, \bar{b}, \bar{c}

\overline{a}+\overline{b}+\overline{c}=\overline{0},|\overline{a}|=3,|\overline{b}|=4,|\overline{c}|=5

\overline{a} \cdot \overline{b}+\overline{b} \cdot \overline{c}+\overline{c} \cdot \overline{a}=

Solution:

Concept:

Squaring the vector sum zero gives the relation between the magnitudes and the pairwise dot products.

Explanation:

Given

\bar{a}+\bar{b}+\bar{c}=\bar{0}

Squaring both sides:

(\bar{a}+\bar{b}+\bar{c})^{2}=|\bar{a}|^{2}+|\bar{b}|^{2}+|\bar{c}|^{2}+2(\bar{a}\cdot\bar{b}+\bar{b}\cdot\bar{c}+\bar{c}\cdot\bar{a})

Since the left side is

\bar{0}^{2}=0

0=3^{2}+4^{2}+5^{2}+2(\bar{a}\cdot\bar{b}+\bar{b}\cdot\bar{c}+\bar{c}\cdot\bar{a})

0=9+16+25+2(\bar{a}\cdot\bar{b}+\bar{b}\cdot\bar{c}+\bar{c}\cdot\bar{a})

0=50+2(\bar{a}\cdot\bar{b}+\bar{b}\cdot\bar{c}+\bar{c}\cdot\bar{a})

2(\bar{a}\cdot\bar{b}+\bar{b}\cdot\bar{c}+\bar{c}\cdot\bar{a})=-50

\bar{a}\cdot\bar{b}+\bar{b}\cdot\bar{c}+\bar{c}\cdot\bar{a}=-25

Answer:

\bar{a}\cdot\bar{b}+\bar{b}\cdot\bar{c}+\bar{c}\cdot\bar{a}=-25

Therefore, the correct option is B.

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