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Applications of Dot Product

a, b, c are n...

Question

a, b, c are non-zero vectors and a is perpendicular to both b and c. If $| a | = 2, | b | = 3, | c | = 4$ and $(b, c) = \frac{2 π}{3}$ , then find $| [a b c] |$ .

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Solution

$∣ ∣ \to a ∣ ∣ = 2 ∣ ∣ ∣ \to b ∣ ∣ ∣ = 3 ∣ ∣ \to c ∣ ∣ = 4$
$∣ ∣ ∣ \to a \to b \to c ∣ ∣ ∣ = \to a . (\to b \times \to c) \to a ⊥ \to b & \to a ⊥ \to c$
$= ∣ ∣ \to a ∣ ∣ ∣ ∣ ∣ \to b \times \to c ∣ ∣ ∣ cos 0^{0}$
$= ∣ ∣ \to a ∣ ∣ ∣ ∣ ∣ \to b \times \to c ∣ ∣ ∣$
$\begin{matrix} = ∣ ∣ \to a ∣ ∣ ∣ ∣ ∣ \to b ∣ ∣ ∣ ∣ ∣ \to c ∣ ∣ sin \frac{2 π}{3} = 2 \times 3 \times 4 \times \frac{\sqrt{3}}{2} \end{matrix}$
$= 12 \sqrt{3}$
Then,
We get $∣ ∣ ∣ [\to a \to b \to c] ∣ ∣ ∣ = 12 \sqrt{3}$

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Q. If the vector a is perpendicular to b and c, |a|=2, |b|=3, |c|=4 and the angle between b and c is

\frac{2 π}{3}

then |[a b c ]| =

Q. If the vector a is perpendicular to b and c, |a|=2, |b|=3, |c|=4 and the angle between b and c is

\frac{2 π}{3}

then |[a b c ]| =

\to a

is perpendicular to

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∣ ∣ \to a ∣ ∣ = 2, ∣ ∣ ∣ \to b ∣ ∣ ∣ = 3, ∣ ∣ \to c ∣ ∣ = 4

and the angle between

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\to c

\frac{2 π}{3}

[\to a \to b \to c]

¯ ¯ ¯ a

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∣ ∣ ¯ ¯ ¯ a ∣ ∣ = 2, ∣ ∣ ¯ ¯ b ∣ ∣ = 3, ∣ ∣ ¯ ¯ c ∣ ∣ = 4

¯ ¯ c . (¯ ¯ ¯ a \times ¯ ¯ b) =

a = a_{1} i + a_{2} j + a_{3} k, b = b_{1} i + b_{2} j + b_{3} k,

c = c_{1} i + c_{2} j + c_{3} k

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{∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2}

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