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Direction Cosines

In a parallel...

Question

In a parallelogram OABC, vectors $\to a, \to b, \to c$ are respectively the position vectors of vertices A, B, C with reference to O is origin. A point $\to E$ is taken on the side BC which divides it in the ratio of 2 : 1. Also, the line segment AE intersects the line bisecting the angel O internally in point P. If CP, when extended meets AB in point F. Then

The position vector of point F is

$\to a + \frac{1}{3} \frac{| \to a |}{| \to c |} \to c$

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$\to a + \frac{2 | \to a |}{| \to c |} \to c$

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$\to a + \frac{1}{2} \frac{| \to [a] |}{| \to c |} \to c$

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$\to a + \frac{| \to a |}{| \to c |} \to c$

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Solution

The correct option is A
$\to a + \frac{1}{3} \frac{| \to a |}{| \to c |} \to c$

$_{1} = \to c + λ_{1} [{\frac{3 | \to a | | \to c |}{3 | \to c | + 2 | \to a |}} {\frac{\to a}{| \to a |} + \frac{\to c}{| \to c |}} - \to c] . . . . . (i)$

Vector equation of line $A B \Rightarrow \to r = \to a + λ_{2} \to c . . . . . . . . . . (i i)$

These two lines intersect at F, we must have for F.

$\to c + λ_{1} [{\frac{3 | \to a | | \to c |}{3 | \to c | + 2 | \to a |}} {\frac{\to a}{| \to a |} + \frac{\to c}{| \to c |}} - \to c] = \to a + λ_{2} \to c$

$\Rightarrow [\frac{3 | \to a | | \to c |}{3 | \to c | + 2 | \to a |} . \frac{1}{| \to a |} - 1] \to a + [1 + \frac{3 λ_{1} | \to a |}{3 | \to c | + 2 | \to a |} - λ_{1} - λ_{2}] \to c = 0$

$\Rightarrow \frac{3 | \to c | λ_{1}}{3 | \to c | + 2 | \to a |} - 1 = 0$

and $1 + \frac{3 λ_{1} | \to a |}{3 | \to c | + 2 | \to a |} - λ_{1} - λ_{2} = 0 [∵ \to a a n d \to c are non-collinear]$

$\Rightarrow λ_{1} = \frac{3 | \to c | + 2 | \to a |}{3 | \to c |} a n d λ_{2} = \frac{1}{3} \frac{| \to a |}{| \to c |}$

Putting value of $λ_{1} i n (i i), we get \to r = \to a + \frac{1}{3} \frac{| \to a |}{| \to c |} \to c$

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In a parallelogram OABC, vectors $\to a, \to b, \to c$ are respectively the position vectors of vertices A, B, C with reference to O is origin. A point $\to E$ is taken on the side BC which divides it in the ratio of 2 : 1. Also, the line segment AE intersects the line bisecting the angel O internally in point P. If CP, when extended meets AB in point F. Then

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