Question

Write four forms of dot product for two vectors A and B by defining their

metric tensor?​

Answer 1

 Here it is in Metric Tensor Form      $g_{\mu v} a^{v} b^{\mu}$

Explanation:

In algebraically, the scalar product is that the sum of the products of the corresponding entries of the 2 sequences of numbers.

In geometrically, it's the merchandise of the Euclidean magnitudes of the 2 vectors and therefore the cosine of the angle between them.

These definitions are equivalent when using Cartesian coordinates.

The scalar product tells you what amount of 1 vector goes within the direction of another.

Therefore the scalar product during this case would offer you the quantity of force getting into the direction of the displacement, or within the direction that the box moved.

In Metric Tensor Form

$a . b = a_{\mu} b^{\mu} = g_{\mu v} a^{v} b^{\mu} = g_{11 } a^{1} b^{1} +g_{12} a^{1} b^{2}+g_{21} a^{2} b^{1}+g_{22} a^{2} b^{2}$