Let a-b-c be three vectors such that - a-b-c-4 and angle between a and b is -3- angle between b and c is -3 and angle between c and a is -3-The volume in cubic units of tetrahedron whose adjacent edges are represented by the vectors a- bandc is

Name: JEE- Grade 12- Mathematics- Vector Algebra- Session 09- W13
Uploaded: 2022-07-04T10:36:42+05:30
Description: Volume of the parallelopiped =| [a b c]| Let |[a b c]| = k [a b c]^2 = a_1 amp;a_2 amp;a_3 b_1 amp;b_2 amp;b_3 c_1 amp;c_2 amp;c_3 a_1 a ...

Volume of the parallelopiped =| [a b c]| Let |[a b c]| = k [a b c]^2 = a_1 amp;a_2 amp;a_3 b_1 amp;b_2 amp;b_3 c_1 amp;c_2 amp;c_3 a_1 a ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard IX

Mathematics

Triangle Inequality

Let a,b,c be ...

Question

Let $\to a, \to b, \to c$ be three vectors such that $| \to a | = | \to b | = | \to c | = 4$ and angle between $\to a$ and $\to b$ is $π / 3,$ angle between $\to b$ and $\to c$ is $π / 3$ and angle between $\to c$ and $\to a$ is $π / 3.$
The volume (in cubic units) of tetrahedron whose adjacent edges are represented by the vectors $\to a, \to b$ and $\to c$ is

\frac{8 \sqrt{2}}{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{16 \sqrt{2}}{3}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{16}{\sqrt{3}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{4 \sqrt{3}}{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $\frac{16 \sqrt{2}}{3}$
Volume of the parallelopiped $= ∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣$
Let $∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣ = k$
$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$
$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a \cdot \to a & \to a \cdot \to b & \to a \cdot \to c \to b \cdot \to a & \to b \cdot \to b & \to b \cdot \to c \to c \cdot \to a & \to c \cdot \to b & \to c \cdot \to c \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$k^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} | \to a |^{2} & | \to a | | \to b | cos π / 3 & | \to a | | \to c | cos π / 3 | \to b | | \to a | cos π / 3 & | \to b |^{2} & | \to b | | \to c | cos π / 3 | \to c | | \to a | cos π / 3 & | \to c | | \to b | cos π / 3 & | \to c |^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$k^{2} = ∣ ∣ ∣ ∣ \begin{matrix} 16 & 8 & 8 8 & 16 & 8 8 & 8 & 16 \end{matrix} ∣ ∣ ∣ ∣$
$k^{2} = 8^{3} ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 1 1 & 2 & 1 1 & 1 & 2 \end{matrix} ∣ ∣ ∣ ∣ = 8^{3} \times 4$
$k = 32 \sqrt{2}$
Hence, volume of parallelopiped $= ∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣ = 32 \sqrt{2}$
Volume of tetrahedron $= \frac{1}{6} ∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣ = \frac{1}{6} \times 32 \sqrt{2}$
Volume of tetrahedron $= \frac{16 \sqrt{2}}{3}$ cubic units.

Suggest Corrections

Similar questions

Q. Let

\to a, \to b, \to c

be three vectors such that

| \to a | = | \to b | = | \to c | = 4

and angle between

\to a

	Column - I		Column - II
(A)	If $\to a, \to b, \to c$ represents adjacent edges of parallelopiped then its volume is	(p)	$\frac{2 \sqrt{2}}{\sqrt{3}}$
(B)	If $\to a, \to b, \to c$ represents adjacent edges of parallelopiped then its height is	(q)	$\frac{2 \sqrt{2}}{3}$
(C)	If $\to a, \to b, \to c$ represents adjacent edges of tetrahedron then its volume is	(r)	$4 \sqrt{2}$
(D)	If $\to a, \to b, \to c$ represents adjacent edges of tetrahedron then its height is	(s)	$\sqrt{\frac{2}{3}}$