If a-b-c- be three non coplanar unit vectors- c- is inclined at an angle - to the plane of a- and b- and a- and c- are inclined at an angle - to b- then a-b- b-c- c-a- is

The correct option is B 12sin 2β sin α(c⃗ a⃗ ) (a×b)×(b×c)×(c×a)=[abc]b×(c×a ) =[abc]{(b. a)c (b. c)a} ... (1) But [abc]=|a×b|c|cos ...

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Byju's Answer

Standard XII

Physics

Horizontal and Vertical Components of Projectile Motion

If a⃗,b⃗,c⃗...

Question

If $\to a, \to b, \to c$ be three non coplanar unit vectors, $\to c$ is inclined at an angle $α$ to the plane of $\to a$ and $\to b$ and $\to a$ and $\to c$ are inclined at an angle $β$ to $\to b$ , then $(\to a \times \to b) \times (\to b \times \to c) \times (\to c \times \to a)$ is

\frac{1}{2} sin 2 α sin β (\to c - \to a)

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\frac{1}{2} sin 2 β sin α (\to c - \to a)

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\frac{1}{2} {sin}^{2} α cos β (\to c - \to a)

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\frac{1}{2} {sin}^{2} β cos α (\to c - \to a)

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Solution

The correct option is B $\frac{1}{2} sin 2 β sin α (\to c - \to a)$
$(¯ ¯ ¯ a \times ¯ ¯ b) \times (¯ ¯ b \times ¯ ¯ c) \times (¯ ¯ c \times ¯ ¯ ¯ a) = [¯ ¯ ¯ a ¯ ¯ b ¯ ¯ c] ¯ ¯ b \times (¯ ¯ c \times ¯ ¯ ¯ a$ )
$= [¯ ¯ ¯ a ¯ ¯ b ¯ ¯ c] {(¯ ¯ b . ¯ ¯ ¯ a) ¯ ¯ c - (¯ ¯ b . ¯ ¯ c) ¯ ¯ ¯ a}$ ... $(1)$
But $[¯ ¯ ¯ a ¯ ¯ b ¯ ¯ c] = | ¯ ¯ ¯ a \times ¯ ¯ b | ¯ ¯ c | cos$ of angle between $(¯ ¯ ¯ a \times ¯ ¯ b)$ and $¯ ¯ c$
$= | ¯ ¯ ¯ a | ¯ ¯ b | sin β | ¯ ¯ c |$ ( $cos$ of angle between the normal to the plane and the line)
$= 1.1. sin β .1$ (sine of angle between the plane and the line )
$= sin β s i n α$
$∴ (¯ ¯ ¯ a \times ¯ ¯ b) \times (¯ ¯ b \times ¯ ¯ c) \times (¯ ¯ c \times ¯ ¯ ¯ a) = sin β sin α {1 .1 cos β [¯ ¯ c - ¯ ¯¯ ¯ a]}$
$= sin β cos β sin α (¯ ¯ c - ¯ ¯ ¯ a)$
$= \frac{1}{2} sin 2 β sin α (¯ ¯ c - ¯ ¯ ¯ a)$

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