In a triangle PQR- let a-QR- b-RP- c-PQ- If-a-3- -b-4 and a-c-bc-a-b-a-a-b-Then the value of -a-b-2 is

For vectors to represents the side of triangle a+b+c=0 ⇒c= a b Now, nbsp;a.(c b)c.(a b)=|a||a|+|b| ⇒a.( a 2b)( a b).(a b)=33+4 ⇒a.(a+2b)(a+b).(a b)=37 ⇒| ...

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Byju's Answer

Standard IX

Mathematics

Triangle Inequality

In a triangle...

Question

In a triangle $P Q R$ , let $\to a = - - \to Q R, \to b = - - \to R P, \to c = - - \to P Q .$ If
$| \to a | = 3,$ $| \to b | = 4$ and $\frac{\to a . (\to c - \to b)}{\to c . (\to a - \to b)} = \frac{| \to a |}{| \to a | + | \to b |}$
Then the value of $| \to a \times \to b |^{2}$ is

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Solution

For vectors to represents the side of triangle $\to a + \to b + \to c = 0$

$\Rightarrow \to c = - \to a - \to b$

Now, $\frac{\to a . (\to c - \to b)}{\to c . (\to a - \to b)} = \frac{| \to a |}{| \to a | + | \to b |}$

$\Rightarrow \frac{\to a . (- \to a - \to 2 b)}{(- \to a - \to b) . (\to a - \to b)} = \frac{3}{3 + 4}$

$\Rightarrow \frac{\to a . (\to a + \to 2 b)}{(\to a + \to b) . (\to a - \to b)} = \frac{3}{7}$

$\Rightarrow \frac{| \to a |^{2} + 2 \to a . \to b}{| \to a |^{2} - | \to b |^{2}} = \frac{3}{7}$

$\Rightarrow 2 \to a . \to b = - 12$

$\Rightarrow \to a . \to b = - 6$

Now, $| \to a \times \to b |^{2} = a^{2} b^{2} - (\to a . \to b)^{2}$

$= 9 \times 16 - (- 6)^{2}$

$= 108$

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Similar questions

Q. If

[\to a \times \to b \to b \times \to c \to c \times \to a] = λ [\to a \to b \to c]^{2}

, then

λ

is equal to

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\to a, \to b

and

\to c

, prove that vectors

\to a - \to b, \to b - \to c

and

\to c - \to a

are coplanar.

\to A

and

\to B

represent force acting on a body in Newton.

\to A

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Q. If

\to a, \to b, \to c

are non - coplanar vectors, then

\frac{a . (b \times c)}{(c \times a) . b} + \frac{b . (a \times c)}{c . (a \times b)} = ?

Q. If

\to a, \to b, \to c, \to d

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[\to a \times \to b \to a \times \to c \to d]

can be simplified as:

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In a triangle PQR, let →a=−−→QR,→b=−−→RP,→c=−−→PQ. If |→a|=3, |→b|=4 and →a.(→c−→b)→c.(→a−→b)=|→a||→a|+|→b| Then the value of |→a×→b|2 is

In a triangle $P Q R$ , let $\to a = - - \to Q R, \to b = - - \to R P, \to c = - - \to P Q .$ If
$| \to a | = 3,$ $| \to b | = 4$ and $\frac{\to a . (\to c - \to b)}{\to c . (\to a - \to b)} = \frac{| \to a |}{| \to a | + | \to b |}$
Then the value of $| \to a \times \to b |^{2}$ is