Vector Algebra Question 145
Question: If a, b and c are unit vectors, then $ |\mathbf{a}-\mathbf{b}{{|}^{2}}+|\mathbf{b}-\mathbf{c}{{|}^{2}}+|\mathbf{c}-\mathbf{a}{{|}^{2}} $ does not exceed
[IIT Screening 2001]
Options:
A) 4
B) 9
C) 8
D) 6
Show Answer
Answer:
Correct Answer: B
Solution:
- $ |\mathbf{a}-\mathbf{b}{{|}^{2}}+|\mathbf{b}-\mathbf{c}{{|}^{2}}+|\mathbf{c}-\mathbf{a}{{|}^{2}} $
$ =2({{\mathbf{a}}^{2}}+{{\mathbf{b}}^{2}}+{{\mathbf{c}}^{2}})-2(\mathbf{a}.\mathbf{b}+\mathbf{b},.\mathbf{c}+\mathbf{c},.,\mathbf{a}) $
$ =2\times 3-2(\mathbf{a},.,\mathbf{b}+\mathbf{b},.,\mathbf{c}+\mathbf{c},.,\mathbf{a}) $
$ =6-{{{(\mathbf{a}+\mathbf{b}+\mathbf{c})}^{2}}-{{\mathbf{a}}^{2}}-{{\mathbf{b}}^{2}}-{{\mathbf{c}}^{2}}} $ $ =9-|\mathbf{a}+\mathbf{b}+\mathbf{c}{{|}^{2}}\le 9. $
BETA
