Q. 6 - Calculus

Question

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

Step-by-Step Solution

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 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

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