The dot product

Calculate dot products and connect them with lengths, angles, and perpendicularity.

Meaning

The dot product produces a scalar from two vectors. In Euclidean coordinates it sums products of corresponding components; for nonzero vectors it equals the product of magnitudes times the cosine of their angle. Compute the squared distance between the vector endpoints both by the cosine rule and by expanding components. Comparing gives the sum of corresponding-component products as |u||v|cosθ. For two nonzero vectors, a zero dot product means cosθ=0, hence a right angle.

Example

so these vectors are perpendicular. Their directional relationship is captured by one number.Before inferring a right angle from a zero dot product, check both vectors are nonzero. The zero vector has no defined direction or angle.

Practice for this concept

Scope and references

Topics

Dot products, angles between vectors, perpendicularity, length calculations

Learning goal

Calculate dot products and connect them with lengths, angles, and perpendicularity.

Helpful background

Explore components, lengths, and angles between directions.

Dot products

These ordinary choice questions provide limited practice in this concept. They do not establish independent solution, full-concept performance, or mastery.

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