SATHEE: Vectors 1 Question 11

Vectors 1 Question 11

12. If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero, non-coplanar vectors and ${\vec{b}}_{1} = \vec{b} - \frac{\vec{b} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a}, \vec{b_{2}} = \vec{b} + \frac{\vec{b} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a}$ $\vec{c_{1}} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a} - \frac{\vec{c} \cdot \vec{b}}{| \vec{b} |^{2}} \vec{b}, \vec{c_{2}} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a} - \frac{\vec{c} \cdot \vec{b_{1}}}{| \vec{b} |^{2}} \vec{b_{1}}$ , $\vec{c_{3}} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a} - \frac{\vec{c} \cdot {\vec{b}}_{2}}{{| {\vec{b}}_{2} |}^{2}} \vec{b_{2}}, \vec{c_{4}} = \vec{a} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a}$

Then, which of the following is a set of mutually orthogonal vectors?

(2005, 1M)

(a) $\vec{a}, {\vec{b}}_{1}, {\vec{c}}_{1}$

(b) $\vec{a}, \vec{b_{1}}, \vec{c_{2}}$

(d) $\vec{a}, {\vec{b}}_{2}, {\vec{c}}_{4}$

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Answer:

Correct Answer: 12. (b)

Solution:

Since, $\vec{b_{1}} = \vec{b} - \frac{\vec{b} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a}, \vec{b_{1}} = \vec{b} + \frac{\vec{b} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a}$

and $\vec{c_{1}} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a} - \frac{\vec{c} \cdot \vec{b}}{| \vec{b} |^{2}} \vec{b} \vec{c_{2}} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a} - \frac{\vec{c} \cdot \vec{b_{1}}}{| \vec{b} |^{2}} \vec{b_{1}}$

$\vec{c_{3}} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a} - \frac{\vec{c} \cdot \vec{b_{2}}}{{| {\vec{b}}_{2} |}^{2}} \vec{b_{2}}, \vec{c_{4}} = \vec{a} - \frac{\vec{c} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a}$

which shows $\vec{a} \cdot \vec{b_{1}} = 0 = \vec{a} \cdot \vec{c_{2}} = \vec{b_{1}} \cdot \vec{c_{2}}$

So, $\vec{a}, {\vec{b}}_{1}, {\vec{c}}_{2}$ are mutually orthogonal vectors.

Vectors 1 Question 11

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Vectors 1 Question 11

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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