If a -3 -4 k-b -2 -2 k-and c- is directed along the direction parallel to angle bisector of a- and b- and the projection of c- on d- is of magnitude 10-9 where d-4 -4 - 7 k- then - C -2 isA- 81B- 120C- 108D- 150

The correct option is B 120direction of angle bisector of nbsp;a and nbsp;b can be calculated as a|a|+b|b| (because resultant of unit vector will lie in di ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

If a =3 ĵ-4 k...

Question

If $\to a = 3^j - 4^k$
$\to b = 2^i -^j + 2^k$
and $\to c$ is directed along the direction parallel to angle bisector of $\to a$ and $\to b$ and the projection of $\to c$ on $\to d$ is of magnitude $\frac{10}{9}$ where $\to d = 4^i - 4^j + 7^k$ then $| \to c |^{2}$ is

81

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120

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108

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150

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Solution

The correct option is B $120$
direction of angle bisector of $\to a$ and $\to b$ can be calculated as
$\frac{\to a}{| \to a |} + \frac{\to b}{| \to b |}$
(because resultant of unit vector will lie in direction parallel to angle bisector)
$= \frac{3^j - 4^k}{\sqrt{(3)^{2} + (- 4)^{2}}} + \frac{2^i -^j + 2^k}{\sqrt{(2)^{2} + (- 1)^{2} + (2)^{2}}}$
$= \frac{3^j - 4^k}{5} + \frac{2^i -^j + 2^k}{3}$
$= \frac{10^i + 4^j - 2^k}{15}$
$\to c = λ \times \frac{10^i + 4^j - 2^k}{15}$
and also
$\to c . \frac{\to d}{| \to d |} = \frac{10}{9}$
$\Rightarrow (λ \times \frac{10^i + 4^j - 2^k}{15}) . (\frac{4^i - 4^j + 7^k}{\sqrt{(4)^{2} + (- 4)^{2} + (7)^{2}}}) = \frac{10}{9}$
$\Rightarrow λ \times \frac{(40 - 16 - 14)}{15 \times 9} = \frac{10}{9}$
$\Rightarrow λ = 15$
$\to c = 15 \times \frac{10^i + 4^j - 2^k}{15}$
$\to c = 10^i + 4^j - 2^k$
$| \to c |^{2} = 120$

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If →a=3^j−4^k →b=2^i−^j+2^k and →c is directed along the direction parallel to angle bisector of →a and →b and the projection of →c on →d is of magnitude 109 where →d=4^i−4^j+7^k then |→c|2 is

If $\to a = 3^j - 4^k$
$\to b = 2^i -^j + 2^k$
and $\to c$ is directed along the direction parallel to angle bisector of $\to a$ and $\to b$ and the projection of $\to c$ on $\to d$ is of magnitude $\frac{10}{9}$ where $\to d = 4^i - 4^j + 7^k$ then $| \to c |^{2}$ is