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Angle between Two Lines

Let a, b, c...

Question

Let $\to a, \to b, \to c$ are the three vectors such that $| \to a |, = | \to b | = | \to c | = 2$ and angle between $\to a$ and $\to b$ is $\frac{π}{3}$ , $\to b$ and $\to c$ is $\frac{π}{3}$ and $\to a$ and $\to c$ is $\frac{π}{3}$ . Now Match the following
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) $\sqrt{\frac{2}{3}}$

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Solution

Given
$| \to a | = | \to c | = | \to c | = 2$
Angle between $\to a$ and $\to b$ is $\frac{π}{3}$
$\to a \cdot \to b = | \to a | ∣ ∣ \to b ∣ ∣ cos (\frac{π}{3})$
$\to a \cdot \to b = 2 \times 2 \times \frac{1}{2}$
$\to a \cdot \to b = 2$

Angle between $\to b$ and $\to c$ is $\frac{π}{3}$
$\to b \cdot \to c = ∣ ∣ \to b ∣ ∣ | \to c | cos (\frac{π}{3})$
$\to b \cdot \to c = 2 \times 2 \times \frac{1}{2}$
$\to b \cdot \to c = 2$

Angle between $\to a$ and $\to c$ is $\frac{π}{3}$
$\to a \cdot \to c = | \to a | | \to c | cos (\frac{π}{3})$
$\to a \cdot \to c = 2 \times 2 \times \frac{1}{2}$
$\to a \cdot \to c = 2$

$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a \cdot \to a & \to a \cdot \to b & \to a \cdot \to c \to b \cdot \to a & \to b \cdot \to b & \to b \cdot \to c \to c \cdot \to a & \to c \cdot \to b & \to c \cdot \to c \end{matrix} ∣ ∣ ∣ ∣ ∣$

$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} {| \to a |}^{2} & \to a \cdot \to b & \to a \cdot \to c \to b \cdot \to a & {∣ ∣ \to b ∣ ∣}^{2} & \to b \cdot \to c \to c \cdot \to a & \to c \cdot \to b & {| \to c |}^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$

$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ \begin{matrix} 4 & 2 & 2 2 & 4 & 2 2 & 2 & 4 \end{matrix} ∣ ∣ ∣ ∣$
$[\to a \to b \to c]^{2} = 4 (16 - 4) - 2 (8 - 4) + 2 (4 - 8)$
$[\to a \to b \to c]^{2} = 48 - 8 - 8$
$[\to a \to b \to c]^{2} = 32$
$[\to a \to b \to c] = \sqrt{32}$
$[\to a \to b \to c] = 4 \sqrt{2}$

(A) volume of parallelopiped
$v o l u m e = [\to a \to b \to c]$
$v o l u m e = 4 \sqrt{2}$
Hence (r) is correct matching

(B) height of parallelopiped
$v o l u m e = \to a \cdot (\to b \times \to c)$

$4 \sqrt{2} = (\to a \cdot^n) ∣ ∣ \to b ∣ ∣ | \to c | sin (\frac{π}{3})$

$4 \sqrt{2} = h e i g h t \times 2 \times 2 \times \frac{\sqrt{3}}{2}$

$4 \sqrt{2} = 2 \sqrt{3} h e i g h t$

$\frac{2 \sqrt{2}}{3} = h e i g h t$
Hence (p) is correct matching

(C) volume of tetrahedron
$v o l u m e = \frac{[\to a \to b \to c]}{6}$

$v o l u m e = \frac{4 \sqrt{2}}{6}$

$v o l u m e = \frac{2 \sqrt{2}}{3}$
Hence (q) is correct matching

(D)height of tetrahedron
$v o l u m e = \frac{1}{3} (A r e a o f b a s e) (H e i g h t)$

$\frac{2 \sqrt{2}}{3} = \frac{1}{3} (∣ ∣ \to b ∣ ∣ | \to c | sin (\frac{π}{3}) (H e i g h t)$

$\frac{2 \sqrt{2}}{3} = \frac{h e i g h t 2 \sqrt{3}}{3}$

$h e i g h t = \frac{\sqrt{2}}{\sqrt{3}}$
Hence (s) is correct matching

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