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Rotation Concept

The volume of...

Question

The volume of tetrahedron with coterminus edges $\to a, \to b$ and $\to c$ is:

A

\frac{1}{6} [\to a \to b \to c]

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B

[\to a \to b \to c]

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C

\to a \times \to b \times \to c

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D

\to a \times \to b . \to c

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Solution

The correct option is A $\frac{1}{6} [\to a \to b \to c]$
A is in another plane (i.e different plane from C, D and B)
Base area $=$ area of triangle BCD
$= \frac{1}{2} ∣ ∣ \to a \times \to b ∣ ∣$
Height, $h = | \to c | cos θ$
AE is the perpendicular from point Alying above plane BCD on the plane BCD
For tetrahedron,
Volume $= \frac{1}{2} \times$ Base area $\times$ height
$= \frac{1}{2} ∣ ∣ \to a \times \to b ∣ ∣ \times | \to c | cos θ \times \frac{1}{3}$
$= \frac{1}{6} ∣ ∣ \to a \times \to b ∣ ∣ \times | \to c | cos θ$
We know $\to p . \to q = | \to p | . | \to q | cos θ$
Let $\to p = \to a \times \to b$
and $\to q = \to c$
$\Rightarrow ∣ ∣ \to a \times \to b ∣ ∣ \times | \to c | cos θ = (\to a \times \to b) \times \to c$
Also, $[\to a \to b \to c] = \to a . (\to b \times \to c)$
and, in $\to a, \to b, \to c$ , on making odd interchange in order, we get a value of same magnitude with negative sign and on making even interchange, we get the same value
$\Rightarrow [\to a \to b \to c] = + [\to c \to a \to b]$
$\Rightarrow \to a . (\to b \times \to c) = \to c . (\to a \times \to b)$
$= (\to a \times \to b) . \to c$
( Since in case of dot product, $\to p . \to q = \to q . \to p$ )
$\Rightarrow$ Volume $= \frac{1}{6} ∣ ∣ \to a \times \to b ∣ ∣ . | \to c |$
$= \frac{1}{6} [\to a \to b \to c]$

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Similar questions

\to a, \to b

\to c

be three non-coplanar vectors and let

\to p, \to q

\to r

be the vectors defined by

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}, \to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

(\to a + \to b) \cdot \to p + (\to b + \to c) \cdot \to q + (\to c + \to a) \cdot \to r =

\to a \times (\to b \times \to c) = \frac{\to b}{3} + \frac{\to c}{2}

\to b \times (\to c \times \to a) = - \frac{\to c}{2}

\to a, \to b

\to c

are non-collinear pair wise unit vectors, then volume of a parallelopiped, whose coterminous edges are

\to a, \to b

\to c

\to a =^i +^j +^k, \to b =^j -^k

, then find a vector

\to c

\to a \times \to c = \to b a n d \to a . \to c = 3

V

is the volume of the parallelopiped having three coterminus edges as

\to a, \to b

\to c

then the volume of the parallelopiped having three coterminus edges as

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

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

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

\to a, \to b

\to c

are perpendicular to

\to b + \to c, \to c + \to a

\to a + \to b

respectively and if

| \to a + \to b | = 6, | \to b + \to c | = 8

| \to c + \to a | = 10

| \to a + \to b + \to c |

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