Question

If a and b are unit vectors in R³, show that |la x b||² +(a∙b)² = 1.

Answer 1

To prove the given identity |la x b||² +(a∙b)² = 1 in R³, we can use the properties of vectors, scalar and vector triple products, and trigonometric identities.

Step-by-step explanation:

To prove the given identity, we can utilize the properties of vectors and the scalar and vector triple products. Let's start by expanding the left-hand side of the equation:

|la x b||² +(a∙b)² = (|a||b|sinθ)² + (a∙b)²

Next, we can use the properties of the scalar and vector triple products to simplify the expression. By applying the triple product identity, we have:

(|a||b|sinθ)² + (a∙b)² = (|a||b|sinθ)² + (|a||b|cosθ)²

Since a and b are unit vectors, |a| = 1 and |b| = 1. So we can simplify further:

(|a||b|sinθ)² + (|a||b|cosθ)² = (sinθ)² + (cosθ)² = 1

Therefore, we have shown that |la x b||² +(a∙b)² = 1.