scalar product and vector product of 4 vectors-pl explain clearly

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Solution

If you mean, "find the scalar product and vector product of 4 vectors", the answer is as follows:-

Let the 4 vectors be $\vec{a} = a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}, \vec{b} = b_{x} \hat{i} + b_{y} \hat{j} + b_{z} \hat{k}, \vec{c} = c_{x} \hat{i} + c_{y} \hat{j} + c_{z} \hat{k}, \vec{d} = d_{x} \hat{i} + d_{y} \hat{j} + d_{z} \hat{k}$ .

Scalar product, S $= \vec{a} . \vec{b} . \vec{c} . \vec{d}$

$\Rightarrow S = (a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}) (b_{x} \hat{i} + b_{y} \hat{j} + b_{z} \hat{k}) (c_{x} \hat{i} + c_{y} \hat{j} + c_{z} \hat{k}) (d_{x} \hat{i} + d_{y} \hat{j} + d_{z} \hat{k})$

$\Rightarrow S = a_{x} b_{x} c_{x} d_{x} + a_{y} b_{y} c_{y} d_{y} + a_{z} b_{z} c_{z} d_{z}$

Vector product, V $= \vec{a} \times \vec{b} \times \vec{c} \times \vec{d}$

$\Rightarrow V = (a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}) \times (b_{x} \hat{i} + b_{y} \hat{j} + b_{z} \hat{k}) \times (c_{x} \hat{i} + c_{y} \hat{j} + c_{z} \hat{k}) \times (d_{x} \hat{i} + d_{y} \hat{j} + d_{z} \hat{k})$

$\Rightarrow V = [(a_{y} b_{z} - a_{z} b_{y}) \hat{i} + (a_{x} b_{z} - a_{z} b_{x}) \hat{j} + (a_{x} b_{y} - a_{y} b_{x}) \hat{k}] \times [(c_{y} d_{z} - c_{z} d_{y}) \hat{i} + (c_{x} d_{z} - c_{z} d_{x}) \hat{j} + (c_{x} d_{y} - c_{y} d_{x}) \hat{k}]$

$\Rightarrow V = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ a_{y} b_{z} - a_{z} b_{y} & a_{x} b_{z} - a_{z} b_{x} & a_{x} b_{y} - a_{y} b_{x} \\ c_{y} d_{z} - c_{z} d_{y} & c_{x} d_{z} - d_{z} b_{x} & c_{x} d_{y} - c_{y} d_{x} \end{matrix}|$

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