SATHEE: Vectors 1 Question 13

Vectors 1 Question 13

14. If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors, then $| \vec{a} - \vec{b} |^{2} + | \vec{b} - \vec{c} |^{2} + | \vec{c} - \vec{a} |^{2}$ does not exceed

(2001, 2M)

(a) 4

(b) 9

(c) 8

(d) 6

Show Answer

Answer:

Correct Answer: 14. (b)

Solution:

Now, $(\vec{a} + \vec{b} + \vec{c})^{2} = Σ {\vec{a}}^{2} + 2 Σ \vec{a} \cdot \vec{b} \geq 0$

$\Rightarrow 2 Σ \vec{a} \cdot \vec{b} \geq - 3$

$[∵ | \vec{a} | = | \vec{b} | = | \vec{c} | = 1]$

Now, $Σ | \vec{a} - \vec{b} |^{2} = 2 Σ {\vec{a}}^{2} - 2 Σ \vec{a} \cdot \vec{b} \leq 2$ (3) $+ 3 = 9$

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