(a⃗ +b⃗ )⋅(a⃗ −b⃗ )

Sure, let's break down the vector operation step by step.

The dot product of two vectors is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. However, in this case, we are given the vectors in terms of a and b, so we can use the properties of the dot product to simplify the expression.

The dot product is distributive over vector addition, so we can expand the expression as follows:

(a + b) . (a - b) = a . a - a . b + b . a - b . b

The dot product of a vector with itself is the square of its magnitude, and the dot product is commutative, meaning a . b = b . a. So we can simplify the expression further:

a . a - a . b + b . a - b . b = |a|^2 - a . b + a . b - |b|^2

The terms a . b and - a . b cancel out, leaving us with:

|a|^2 - |b|^2

So, (a + b) . (a - b) = |a|^2 - |b|^2.