Consider - ABC with A- a-B- b- and C- c- If b- a-c- -b-b-a-c- - b-a- -3- - c-b- -4 then the angle between the medians AM and BD is

The correct option is A π cos ^ 1 ( 1/5√(13) ) b. ( a+c )=b.b+a.c b. ( a b)+c. ( b a )=0 ( b c ). ( a b )=0 ⇒ #160;angle between ...

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Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

Consider Δ ...

Question

Consider $Δ A B C$ with $A \equiv (\to a), B \equiv (\to b)$ and $C = (\to c),$ If $\to b . (\to a + \to c) = \to b . \to b + \to a . \to c; ∣ ∣ \to b - \to a ∣ ∣ = 3; ∣ ∣ \to c - \to b ∣ ∣ = 4$ then the angle between the medians $¯ ¯¯¯¯¯¯¯¯ ¯ A M$ and $¯ ¯¯¯¯¯¯¯ ¯ B D$ is

π - {cos}^{- 1} (\frac{1}{5 \sqrt{13}})

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π - {cos}^{- 1} (\frac{1}{13 \sqrt{5}})

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{cos}^{- 1} (\frac{1}{5 \sqrt{13}})

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{cos}^{- 1} (\frac{1}{13 \sqrt{5}})

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Solution

The correct option is A $π - {cos}^{- 1} (\frac{1}{5 \sqrt{13}})$

$\to b . (\to a + \to c) = \to b . \to b + \to a . \to c$

$\to b . (\to a - \to b) + \to c . (\to b - \to a) = 0$

$(\to b - \to c) . (\to a - \to b) = 0$

$\Rightarrow$ angle between $\to b - \to c$ & $\to a - \to b$
Let B be the origin & $A (3^i), C (4^j)$ then $M (2^j)$ & $D (\frac{3^i + 4^j}{2})$
$∴$ Angle between $B D$ & $A M$

$cos θ = \frac{(2^j - 3^i .) (\frac{3^i + 4^j}{2})}{\sqrt{13} \times \frac{5}{2}} = - \frac{1}{5 \sqrt{13}}$

$θ = π - {cos}^{- 1} \frac{1}{5 \sqrt{13}} .$

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