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 height of the parallelopiped whose adjacent edges are represented by the vectors a- bandc

Name: JEE- Grade 12- Mathematics- Vector Algebra- Session 09- W12
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 ...

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Byju's Answer

Standard XII

Physics

Comparing Vectors

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 height of the parallelopiped whose adjacent edges are represented by the vectors $\to a, \to b$ and $\to c$

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

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

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

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

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Solution

The correct option is A
$4 \sqrt{\frac{2}{3}}$

Volume of the parallelopiped $= ∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣$
Let $∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣ = k$
$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$
$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a . \to a & \to a . \to b & \to a . \to c \to b . \to a & \to b . \to b & \to b . \to c \to c . \to a & \to c . \to b & \to c . \to c \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$k^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} | \to a |^{2} & | \to a | | \to b | cos π / 3 & | \to a | | \to c | cos π / 3 | \to b | | \to a | cos π / 3 & | \to b |^{2} & | \to b | | \to c | cos π / 3 | \to c | | \to a | cos π / 3 & | \to c | | \to b | cos π / 3 & | \to c |^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$k^{2} = ∣ ∣ ∣ ∣ \begin{matrix} 16 & 8 & 8 8 & 16 & 8 8 & 8 & 16 \end{matrix} ∣ ∣ ∣ ∣$
$k^{2} = 8^{3} ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 1 1 & 2 & 1 1 & 1 & 2 \end{matrix} ∣ ∣ ∣ ∣ = 8^{3} \times 4$
$k = 32 \sqrt{2}$
Hence, volume of parallelopiped $= ∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣ = 32 \sqrt{2}$
Height of parallelopiped $= \frac{Volume of parallelopiped}{Base Area}$
Base Area $= | \to a \times \to b | = | \to b \times \to c | = | \to c \times \to a |$
Base Area $= | \to a | | \to b | sin π / 3 = 4 \times 4 \times \frac{\sqrt{3}}{2} = 8 \sqrt{3}$
Height of parallelopiped $= \frac{32 \sqrt{2}}{8 \sqrt{3}} = 4 \sqrt{\frac{2}{3}}$

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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)	$\sqrt{\frac{2}{3}}$

Let →a,→b,→c be three vectors such that |→a|=|→b|=|→c|=4 and angle between →a and →b is π/3, angle between →b and →c is π/3 and angle between →c and →a is π/3. The height of the parallelopiped whose adjacent edges are represented by the vectors →a,→b and →c