Question

What is Lagrange’s identity? How is it used to understand the geometry of the cross product?

Short answer

A relationship between the dot product and cross product and help us understand the geometry of the cross product is known as Lagrange’s identity.

It is used to understand the geometry of the cross product as${||u\times v||}^2={||u||}^2{||v||}^2-{(u·v)}^2$.

Step-by-step solution

Step 1. Given Information 

We have to explain what is Lagrange’s identity and how is it used to understand the geometry of the cross product.

Step 2. The Lagrange’s identity

A relationship between the dot product and cross product and will be used shortly to help us understand the geometry of the cross product. It is known as Lagrange’s identity

Step 3. It is used to understand the geometry of the cross product as

Let u and v be vectors in ${\mathrm{ℝ}}^3$. Then

${||u\times v||}^2={||u||}^2{||v||}^2-{(u·v)}^2$