SATHEE: Vectors 3 Question 28

Vectors 3 Question 28

28. Let $V$ be the volume of the parallelopiped formed by the vectors

$c = c_{1} i + c_{2} \hat{j} + c_{3} k$

If $a_{r}, b_{r}, c_{r}$ , where $r = 1, 2, 3$ are non-negative real numbers and $\sum_{r = 1}^{3} (a_{r} + b_{r} + c_{r}) = 3 L$ . Show that $V \leq L^{3}$ . $(2002, 5 M))$

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Solution:

$V = | \vec{a} \cdot (\vec{b} \times \vec{c}) | \leq \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$

$\sqrt{b_{1}^{2} + b_{2}^{2} + b_{3}^{2}} \sqrt{c_{1}^{2} + c_{2}^{2} + c_{3}^{2}}$

Now, $L = \frac{(a_{1} + a_{2} + a_{3}) + (b_{1} + b_{2} + b_{3}) + (c_{1} + c_{2} + c_{3})}{3}$

$\geq {[(a_{1} + a_{2} + a_{3}) (b_{1} + b_{2} + b_{3}) (c_{1} + c_{2} + c_{3})]}^{1 / 3}$

[using AM $\geq$ GM]

$\Rightarrow L^{3} \geq [(a_{1} + a_{2} + a_{3}) (b_{1} + b_{2} + b_{3}) (c_{1} + c_{2} + c_{3})]$

Now, ${(a_{1} + a_{2} + a_{3})}^{2}$

$= a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + 2 a_{1} a_{2} + 2 a_{1} a_{3} + 2 a_{2} a_{3} \geq a_{1}^{2} + a_{2}^{2} + a_{3}^{2}$

$\Rightarrow (a_{1} + a_{2} + a_{3}) \geq \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$

Similarly, $(b_{1} + b_{2} + b_{3}) \geq \sqrt{b_{1}^{2} + b_{2}^{2} + b_{3}^{2}}$

and $(c_{1} + c_{2} + c_{3}) \geq \sqrt{c_{1}^{2} + c_{2}^{2} + c_{3}^{2}}$

$∴ L^{3} \geq {[(a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) (b_{1}^{2} + b_{2}^{2} + b_{3}^{2}) (c_{1}^{2} + c_{2}^{2} + c_{3}^{2})]}^{1 / 2}$

$\Rightarrow L^{3} \geq V$

[from Eq. (i)]

Vectors 3 Question 28

28. Let $V$ be the volume of the parallelopiped formed by the vectors

Solution:

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Vectors 3 Question 28

28. Let V be the volume of the parallelopiped formed by the vectors

Solution:

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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28. Let $V$ be the volume of the parallelopiped formed by the vectors