Proving Identity: |a × b|² + (a•b)² = |a|²|b|²

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In summary, the given identity relates three different kinds of products and can be proven using the equations for the cross product and dot product of two vectors. The final step involves simplifying the expression using the trigonometric identities sin^2(theta)+cos^2(theta)=1 and the distributive property.
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Homework Statement

The identity below is significant because it relates 3 different kinds of products: a cross product and a dot product of 2 vectors on the left side, and the product of 2 real numbers on the right side. Prove the identity below.

| a × b |² + (a • b)² = |a|²|b|²

Homework Equations

| a × b | = |a||b|sinθ
(a • b) = |a||b|cosθ

The Attempt at a Solution

My work, LSH:

= | a × b |² + (a • b)²

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

= (|a|²)(|a||b|)(|a|sinθ)(|a||b|)(|b|²)(|b|sinθ)(|a| sinθ)(|b|sinθ)(sin²θ) + (|a|²)(|a||b|)(|a|cosθ)(|a||b|)(|b|²)(|b|cosθ)(|a| cosθ)(|b|cosθ)(cos²θ)

= (|a|²)(|a||b|)²(|a|sinθ)²(|b|²)(|b|sinθ)²(sin²θ) + (|a|²)(|a||b|)²(|a|cosθ)²(|b|²)(|b|cosθ)²(cos²θ)

= (|a|²|b|²(|a||b|)²) [(|a|sinθ)²(|b|sinθ)²(sin²θ) + (|a|cosθ)²(|b|²)(|b|cosθ)²(cos²θ)]

= (|a|²|b|²(|a||b|)²) [(|a|²)(sin²θ)(|b|²)(sin²θ)(sin²θ) + (|a|²)(cos²θ)(|b|²)(|b|²)(cos²θ)(cos²θ)]

= (|a|²|b|²(|a||b|)²) [(sin²θ)(sin²θ)(sin²θ) + (cos²θ)(cos²θ)(cos²θ)]

And now I don't know what else to do! Please help. Did I mess up somewhere in my steps? Or is it possible to common factor still?
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  • #2
Stop with this. (|a||b|sinθ)(|a||b|sinθ) + (|a||b|cosθ)(|a||b|cosθ). That's |a|^2*|b|^2*(sin^2(theta)+cos^2(theta)). Now what? I don't know what you are doing on the following lines.

1. What is the purpose of the equation |a × b|² + (a•b)² = |a|²|b|²?

The equation is used to prove the identity of two vectors, a and b, in a vector space. It shows that the magnitude of the cross product of a and b squared, plus the magnitude of the dot product of a and b squared, is equal to the product of the magnitudes of a and b squared.

2. How is the equation |a × b|² + (a•b)² = |a|²|b|² derived?

The equation is derived from the properties of vector operations and the definition of the dot and cross products. It can also be proven using geometric and algebraic methods.

3. What does the equation |a × b|² + (a•b)² = |a|²|b|² represent geometrically?

The equation represents the relationship between the lengths of two vectors, a and b, and the angle between them. It shows that the sum of the squares of the magnitudes of the cross and dot products is equal to the square of the product of the magnitudes of the vectors.

4. How is the equation |a × b|² + (a•b)² = |a|²|b|² used in real-world applications?

The equation is used in various fields of science and engineering, such as physics, mechanics, and computer graphics. It is used to calculate the magnitude of forces, torque, and projections in 3D space.

5. Can the equation |a × b|² + (a•b)² = |a|²|b|² be generalized to higher dimensions?

Yes, the equation can be generalized to higher dimensions and is applicable to any number of vectors. In higher dimensions, the equation becomes the Pythagorean theorem for vector spaces.

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