Let 𝐯 and 𝐰 denote two nonzero vectors. Show that the vectors 𝐰 𝐯+𝐯 𝐰 and 𝐰 𝐯-𝐯 𝐰 are orthogonal

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Let 𝐯 and 𝐰 denote two nonzero vectors. Show that the...

Let $\mathbf{v}$ and $\mathbf{w}$ denote two nonzero vectors. Show that the vectors $\|\mathbf{w}\| \mathbf{v}+\|\mathbf{v}\| \mathbf{w}$ and $\|\mathbf{w}\| \mathbf{v}-\|\mathbf{v}\| \mathbf{w}$ are orthogonal.

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Key Concepts

Orthogonality

Orthogonality means that two vectors are perpendicular to each other, which is characterized by their dot product being zero. This concept is central in linear algebra as it ensures that the vectors do not share any common component in a given direction, simplifying many analyses in geometry and optimization.

Dot Product

The dot product is a fundamental operation in vector algebra that multiplies two vectors to yield a scalar, capturing the idea of projection and angle between the vectors. Its properties, such as commutativity and distributivity over vector addition, are critical in proving results about orthogonality by expanding and simplifying expressions.

Vector Norm

A vector norm is a measure of the length or magnitude of a vector. It is often defined via the square root of the dot product of the vector with itself, and is an essential concept in quantifying distances and scaling in vector spaces.

Distributive Property of the Dot Product

The distributive property allows one to expand the dot product over sums of vectors, which is key in demonstrating orthogonality in proofs. By breaking down complex dot product expressions into simpler components, this property helps in systematically showing that the total dot product is zero.

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