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Scalar Triple Product

Let a=α1î+α2ĵ...

Question

$Let \to a = α_{1}^i + α_{2}^j + α_{3}^k,$ $\to b = β_{1}^i + β_{2}^j + β_{3}^k$ and $\to c = γ_{1}^i + γ_{2}^j + γ_{3}^k,$ $| \to a | = 2 \sqrt{2}, \to a$ makes an angle $\frac{π}{6}$ with the plane of $\to b, \to c$ and $(\to b, \to c) = \frac{π}{3},$ then ${∣ ∣ ∣ ∣ \begin{matrix} α_{1} & α_{2} & α_{3} β_{1} & β_{2} & β_{3} γ_{1} & γ_{2} & γ_{3} \end{matrix} ∣ ∣ ∣ ∣}^{n}$ is equal to
( $n$ is even natural number)

A

{⎛ ⎜ ⎝ \frac{| \to b | | \to a |}{6} ⎞ ⎟ ⎠}^{\frac{n}{2}}

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B

{⎛ ⎜ ⎝ \frac{3 | \to b | | \to c |}{\sqrt{2}} ⎞ ⎟ ⎠}^{n}

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C

\frac{| \to b | | \to c |^{n / 2}}{\sqrt{3} {.2}^{n}}

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D

{⎛ ⎜ ⎝ \frac{\sqrt{2} | \to b | | \to a |}{\sqrt{3}} ⎞ ⎟ ⎠}^{n}

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Solution

The correct option is B ${⎛ ⎜ ⎝ \frac{3 | \to b | | \to c |}{\sqrt{2}} ⎞ ⎟ ⎠}^{n}$
The volume of parallelopiped made by vectors $\to a, \to b$ and $\to c$ is given by $∣ ∣ ∣ ∣ \begin{matrix} α_{1} & α_{2} & α_{3} β_{1} & β_{2} & β_{3} γ_{1} & γ_{2} & γ_{3} \end{matrix} ∣ ∣ ∣ ∣ = | \to a \cdot (\to b \times \to c) | = | \to a | | \to b \times \to c | cos \frac{π}{6} = | \to a | | \to b | | \to c | sin \frac{π}{3} cos \frac{π}{6} = 2 \sqrt{2} | \to b | | \to c | \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} (∵ | \to a | = 2 \sqrt{2}) = \frac{3 | \to b | | \to c |}{\sqrt{2}}$
$∴ {∣ ∣ ∣ ∣ \begin{matrix} α_{1} & α_{2} & α_{3} β_{1} & β_{2} & β_{3} γ_{1} & γ_{2} & γ_{3} \end{matrix} ∣ ∣ ∣ ∣}^{n} = {⎛ ⎜ ⎝ \frac{3 | \to b | | \to c |}{\sqrt{2}} ⎞ ⎟ ⎠}^{n}$

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Similar questions

¯ a = α_{1}^i + α_{2}^j + α_{3}^k

¯ b = β_{1}^i + β_{2}^j + β_{3}^k

¯ c = γ_{1}^i + γ_{2}^j + γ_{3}^k

| ¯ a | = 2 \sqrt{2}

¯ a

\frac{π}{3}

¯ b

¯ c

and angle between

¯ b

¯ c

\frac{π}{3}

{∣ ∣ ∣ ∣ \begin{matrix} α_{1} & α_{2} & α_{3} β_{1} & β_{2} & β_{3} γ_{1} & γ_{2} & γ_{3} \end{matrix} ∣ ∣ ∣ ∣}^{n}

is equal to (n is even natural number).

\to a, \to b, \to c

are unit vectors ,

\to b

\to c

are inclined to

\to a

\frac{π}{6}

and the angle between

\to b

\to c

\frac{π}{3}

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

a, b, c

are the sides opposite to angles

A, B, C

A B C,

respectively and

∠ A = \frac{π}{3}, b : c = \sqrt{3} + 1 : 2,

then the value of

∠ B - ∠ C

\to a, \to b, \to c

are unit vectors equally inclined to each other at an angle

\frac{π}{3}

[\to b + \to c, \to c + \to a, \to a + \to b] =

Q. If a,b,c are non-coplanar unit vectors such that

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

, then the angle between a and b is

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Scalar Triple Product

Standard XII Mathematics

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