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Applications of Cross Product

If ℓ⃗,m⃗,n⃗...

Question

If $\to ℓ, \to m, \to n$ are three non-coplanar vectors and $\to a, \to b$ are two vectors, then prove that
$[\to ℓ, \to m,^n] (\to a, \to b) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to ℓ . \to a & \to ℓ . \to b & \to ℓ \to m . \to a & \to m . \to b & \to m \to n . \to a & \to n . \to b & \to n \end{matrix} ∣ ∣ ∣ ∣ ∣$

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Solution

We have to prove that
$∴ [\to l \to m \to n] (\to a \times \to b) = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to l . \to a & \to l . \to b & \to l . \to c \to m . \to a & \to m . \to b & \to m . \to n \to n . \to a & \to n . \to b & \to n . \to c \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$L e t \to l = l_{1} ˆ i + l_{2} ˆ j + l_{3} ˆ k$
$\to n = n_{1} ˆ i + n_{2} ˆ j + n_{3} ˆ k$
$\to m = m_{1} ˆ i + m_{2} ˆ j + m_{3} ˆ k$
$\to a = a_{1} ˆ i + a_{2} ˆ j + a_{3} ˆ k$ & $\to b = b_{1} ˆ i + b_{2} ˆ j + b_{3} ˆ k$
Now,
$\to l . \to a = l_{1} a_{1} + l_{2} a_{2} + l_{3} a_{3} = \sum l_{1} a_{1}$
Similarly
$\to l . \to b = \sum l_{1} b_{1}$
$N o w$
$[\to l \to m \to n] [\to a \times \to b] = ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & l_{2} & l_{3} m_{1} & m_{2} & m_{1} n_{1} & n_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ \times ∣ ∣ ∣ ∣ ∣ \begin{matrix} ˆ i & ˆ j & ˆ k a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} \sum l_{1} ˆ i & \sum l_{1} a_{1} & \sum l_{1} b_{1} \sum m_{1} ˆ i & \sum m_{1} a_{1} & \sum l_{1} b_{1} \sum n_{1} ˆ i & \sum n_{1} a_{1} & \sum l_{1} b_{1} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to l & \to l . \to a & \to l . \to b \to m & \to m . \to a & \to m . \to b \to n & \to n . \to a & \to n . \to b \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to l . \to a & \to l . \to b & \to l \to m . \to a & \to m . \to b & \to m \to n . \to a & \to n . \to b & \to n \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
Hence, proved.

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