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Test for Coplanarity

Let a,b,c b...

Question

Let $a, b, c$ be three vectors. Then show that
$¯ ¯ ¯ a \times (¯ ¯ b \times ¯ ¯ c) = (¯ ¯ ¯ a . ¯ ¯ c) ¯ ¯ b - (¯ ¯ ¯ a . ¯ ¯ b) ¯ ¯ c$

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Solution

Let $\to r = (\to a \times \to b) \times \to c$
since, cross product of two vectors is a vector perpendicular to both the vectors
$∴ \to r = \to a \times (\to b \times \to c)$ -----(1)
$\Rightarrow \to r ⊥ \to a a n d \to r ⊥ (\to b \times \to c)$
But $(\to b \times \to c)$ is a vector perpendicular to the plane of $\to b$ and $\to c$ therefore
$\to r ⊥ (\to b \times \to c)$
$\Rightarrow \to r$ lies in the plane $\to b$ and $\to c$
$ \Rightarrow \overrightarrow r $ is expressible as linear combination of $\to b$ and $\to c$
$\Rightarrow T h e r e e x i s t s c a l e r x, y$
such that
$\to r = x \to b + y \to c - - - - - (2)$
Now
$\to r ⊥ \to a$
$\Rightarrow \to r . \to a = 0$
$\Rightarrow (x \to b + y \to c) . \to a = 0$

$\Rightarrow x (\to b . \to a) + y (\to c . \to a) = 0$

$\Rightarrow x (\to a . \to b) + y (\to a . \to c) = 0$

$\Rightarrow x (\to a . \to b) = - y (\to a . \to c)$

$\Rightarrow \frac{x}{(\to a . \to c)} = \frac{y}{- (\to a . \to b)} = λ (s a y)$

$\Rightarrow x = λ (\to a . \to c)$
$y = - λ (\to a . \to b)$
Substituting the vector of $x$ and $y$ in (2) we get
$\to r = λ (\to a . \to c) \to b - λ (\to a . \to b) \to c$

$\Rightarrow \to r = λ [(\to a . \to c) \to b - λ (\to a . \to b) \to c]$

$\Rightarrow \to a \times (\to b \times \to c) = λ [(\to a . \to c) \to b - λ (\to a . \to b) \to c]$
This is vector identity and so it is true $\forall \to a, \to b, \to c$
$∴ \to a \times (\to b \times \to c) = (\to a . \to c) \to b - (\to a . \to b) \to c$
Hence, the given question is proved

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Similar questions

a, b, c

be three vectors. Then show that

(¯ ¯ ¯ a \times ¯ ¯ b) \times ¯ ¯ c = (¯ ¯ ¯ a . ¯ ¯ c) ¯ ¯ b - (¯ ¯ b . ¯ ¯ c) ¯ ¯ ¯ a

Q. Vector product of three vectors

\to a, \to b

\to c

\to a \times (\to b \times \to c)

is known as vector triple product. It is defined as

\to a \times (\to b \times \to c) = (\to a . \to c) \to b - (\to a . \to b) \to c

. Vector triple product

\to a \times (\to b \times \to a)

a, b, c

are three non coplanar, non zero vectors, then

(a . a) b \times c + (a . b) c \times a + (a . c) a \times b

¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c

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(¯ ¯ ¯ a . ¯ ¯ ¯ a) (¯ ¯ b \times ¯ ¯ c) + (¯ ¯ ¯ a . ¯ ¯ b) (¯ ¯ c \times ¯ ¯ ¯ a) + (¯ ¯ ¯ a . ¯ ¯ c) (¯ ¯ ¯ a \times ¯ ¯ b)

(\to a . \to b) \to c - (\to a . \to c) \to b

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Test for Coplanarity

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