SCALAR PRODUCT and geometrics

Question

The question is : Geometrically when is $(a\cdot b)^2 = (a^2) (b^2)$?

Do I use index notation or maybe just use the definition of dot product. please help

Answer 1

Geometrically, if $\theta$ is the angle between $a$ and $b$, we have $$\cos\theta=\frac{a\cdot b}{|a||b|}$$ We can square this and we get $$\cos^2\theta=\frac{(a\cdot b)^2}{|a|^2|b|^2}$$ Using $|x|^2=x^2$, and $(a\cdot b)^2=a^2b^2$, we get that $\cos^2\theta=1$. The solution is $\cos\theta=\pm 1$. That means that the vectors are either parallel or antiparallel.

Answer 2

Given two vectors $a$ and $b$, we call $G:=(a)^2(b)^2-(a\cdot b)^2$ the Gram-determinante of these vectors, see https://en.wikipedia.org/wiki/Gramian_matrix#Gram_determinant. The square root of $G$ ist the area of the parallelogram spanned by the two vectors. Now if $G=0$, the vectors must be linearly dependent.