If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that → B C + → C A + → A B = → 0 and deduce that a sin A = b sin B = c sin C . - Mathematics

प्रश्न

If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that $\vec{B C} + \vec{C A} + \vec{A B} = \vec{0}$ and deduce that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .$

योग

उत्तर

$We have$
$\vec{B C} = \vec{a}$
$\vec{C A} = \vec{b}$
$\vec{A B} = \vec{c}$
$| \vec{a} | = a$
$| \vec{b} | = b (∵ Length is always positive)$
$\vec{c} = c$
$Now,$
$\vec{B C} + \vec{C A} + \vec{A B} = \vec{0} (Given)$
$\Rightarrow \vec{a} + \vec{b} + \vec{c} = \vec{0}$
$\Rightarrow \vec{a} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{a} \times \vec{0}$
$\Rightarrow \vec{a} \times \vec{a} + \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = \vec{0}$
$\Rightarrow \vec{0} + \vec{a} \times \vec{b} - \vec{c} \times \vec{a} = \vec{0}$
$\Rightarrow \vec{a} \times \vec{b} = \vec{c} \times \vec{a}$
$\Rightarrow | \vec{a} | | \vec{b} | \sin C = | \vec{c} | | \vec{a} | \sin B$
$\Rightarrow a b \sin C = c a \sin B$
$Dividing both sides by abc,we get$
$\Rightarrow \frac{\sin C}{c} = \frac{\sin B}{b} . . . (1)$
$Again,$
$\vec{B C} + \vec{C A} + \vec{A B} = \vec{0}$
$\Rightarrow \vec{a} + \vec{b} + \vec{c} = \vec{0}$
$\Rightarrow \vec{b} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{b} \times \vec{0}$
$\Rightarrow \vec{b} \times \vec{a} + \vec{b} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0}$
$\Rightarrow - \vec{a} \times \vec{b} + \vec{0} + \vec{b} \times \vec{c} = \vec{0}$
$\Rightarrow \vec{a} \times \vec{b} = \vec{b} \times \vec{c}$
$\Rightarrow | \vec{a} | | \vec{b} | \sin C = | \vec{b} | | \vec{c} | \sin A$
$\Rightarrow a b \sin C = b c \sin A$
$Dividing both sides by abc,we get$
$\Rightarrow \frac{\sin C}{c} = \frac{\sin A}{a} . . . (2)$
$From (1) and (2), we get$
$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$
$\Rightarrow \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

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Product of Two Vectors - Vector (Or Cross) Product of Two Vectors

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Q 20 Q 19 Q 21

अध्याय 25: Vector or Cross Product - Exercise 25.1 [पृष्ठ ३०]

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आरडी शर्मा Mathematics [English] Class 12

अध्याय 25 Vector or Cross Product
Exercise 25.1 | Q 20 | पृष्ठ ३०

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If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that → B C + → C A + → A B = → 0 and deduce that a sin A = b sin B = c sin C . - Mathematics

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