Vectors 1 Question 13
14. If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are unit vectors, then $|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|^{2}+|\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}|^{2}+|\overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}}|^{2}$ does not exceed
(2001, 2M)
(a) 4
(b) 9
(c) 8
(d) 6
Show Answer
Answer:
Correct Answer: 14. $\frac{8}{3}$
Solution:
- Now, $(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})^{2}=\Sigma \overrightarrow{\mathbf{a}}^{2}+2 \Sigma \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}} \geq 0$ $\Rightarrow \quad 2 \Sigma \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}} \geq-3$ $[\because|\overrightarrow{\mathbf{a}}|=|\overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{c}}|=1]$ Now, $\quad \Sigma|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|^{2}=2 \Sigma \overrightarrow{\mathbf{a}}^{2}-2 \Sigma \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}} \leq 2$ (3) $+3=9$
