If a- b- c be three non-coplanar uni-modular vectors each inclined with other at an angle of 60- then volume of the tetrahedron whose edges are a- b and c is

Volume of tetrahedron with edges a, b, c is nbsp; v = 16 [abc] ⇒ v^2 = 136 [abc]^2 = 136a.a amp; a.b amp; a.c b.a amp; b.b amp; b.c c.a amp; c.b a ...

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Standard XII

Mathematics

Applications Scalar Triple Product

If a, b, c be...

Question

If $\to a, \to b, \to c$ be three non-coplanar uni-modular vectors each inclined with other at an angle of $60^{\circ}$ , then volume of the tetrahedron whose edges are $\to a, \to b$ and $\to c$ is

4 \sqrt{2} cubic units

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\frac{1}{6 \sqrt{2}} cubic units

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\frac{1}{\sqrt{2}} cubic units

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6 \sqrt{2} cubic units

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Solution

The correct option is A $4 \sqrt{2} cubic units$
Volume of tetrahedron with edges $\to a, \to b, \to c$ is
$v = \frac{1}{6} [\to a \to b \to c]$
$\Rightarrow v^{2} = \frac{1}{36} [\to a \to b \to c]^{2}$
$= \frac{1}{36} ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a . \to a & \to a . \to b & \to a . \to c \to b . \to a & \to b . \to b & \to b . \to c \to c . \to a & \to c . \to b & \to c . \to c \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$= \frac{1}{36} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \frac{1}{2} & 1 & \frac{1}{2} \frac{1}{2} & \frac{1}{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$= \frac{1}{36} [1 (1 - \frac{1}{4}) - \frac{1}{2} (\frac{1}{2} - \frac{1}{4}) + \frac{1}{2} (\frac{1}{4} - \frac{1}{2})]$
$= \frac{1}{36} [\frac{4 - 1}{4} - \frac{1}{2} (\frac{2 - 1}{4}) + \frac{1}{2} (\frac{1 - 2}{4})]$
$= \frac{1}{36} [\frac{3}{4} - \frac{1}{8} - \frac{1}{8}]$
$= \frac{1}{36} [\frac{3}{4} - \frac{1}{4}] = \frac{1}{36} \times \frac{2}{4} = \frac{1}{36 \times 2}$
$∴ v = \frac{1}{6 \sqrt{2}} cubic unit$

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If →a,→b,→c be three non-coplanar uni-modular vectors each inclined with other at an angle of 60∘, then volume of the tetrahedron whose edges are →a,→b and →c is

If $\to a, \to b, \to c$ be three non-coplanar uni-modular vectors each inclined with other at an angle of $60^{\circ}$ , then volume of the tetrahedron whose edges are $\to a, \to b$ and $\to c$ is