Suppose 𝐮, 𝐯 satisfy 𝐮=3,𝐮+𝐯=4, and 𝐮-𝐯=6. What must 𝐯 equal? Does your answer depend upon which

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Suppose 𝐮, 𝐯 satisfy 𝐮=3,𝐮+𝐯=4, and 𝐮-𝐯=6. What must 𝐯...

Suppose $\mathbf{u}, \mathbf{v}$ satisfy $\|\mathbf{u}\|=3,\|\mathbf{u}+\mathbf{v}\|=4$, and $\|\mathbf{u}-\mathbf{v}\|=6$. What must $\|\mathbf{v}\|$ equal? Does your answer depend upon which norm is being used?

Key Concepts

Dependence on the Underlying Norm

The result obtained using algebraic relationships such as the parallelogram law depends on the norm being used. In spaces where the norm is induced by an inner product (like the Euclidean norm), these relationships hold and yield a unique answer. In contrast, in more general normed spaces where such properties may not hold, the answer might not be determined uniquely.

Parallelogram Law

The parallelogram law states that for any vectors in an inner product space, the sum of the squares of the lengths of the vectors added and subtracted is equal to twice the sum of the squares of the individual vector lengths. This property, which only holds in inner product spaces, is key to solving problems where the norms of vector combinations are known.

Norm

A norm is a function that assigns a non-negative length or size to each vector in a vector space, obeying properties like positivity, homogeneity, and the triangle inequality. Understanding norms is essential in analyzing and comparing the lengths of vectors or their combinations in different vector spaces.

Euclidean (Inner Product) Norm

The Euclidean norm is defined by the square root of the sum of the squares of the components of a vector. It is the standard way of measuring distance in Euclidean spaces and crucially satisfies the parallelogram law, which allows for the derivation of relationships involving the norms of sums and differences of vectors.

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