Find three vectors 𝐮, 𝐯 and 𝐰 in ℝ^3 such that 𝐮 and 𝐯 are orthogonal, 𝐮 and 𝐰 are orthogonal, b

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Find three vectors 𝐮, 𝐯 and 𝐰 in ℝ^3 such that 𝐮 and 𝐯 are...

Find three vectors $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ in $\mathbb{R}^3$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, $\mathbf{u}$ and $\mathbf{w}$ are orthogonal, but $\mathbf{v}$ and $\mathbf{w}$ are not orthogonal. Are your vectors linearly independent or linearly dependent? Can you find vectors of the opposite dependency satisfying the same conditions? Why or why not?

Key Concepts

Linear Dependence in Vector Spaces

Linear dependence occurs when at least one vector in a collection can be expressed as a linear combination of the others. This relationship is crucial when analyzing vector sets in spaces such as ?³ because it highlights whether the set spans the entire space or a subspace thereof. Recognizing the conditions that lead to linear dependence helps in understanding the structure and dimension of subspaces and is key when determining if a set of constraints, such as specific orthogonality conditions, can lead to a fully spanning independent set or a dependent one.

Orthogonality

Orthogonality is a fundamental concept in linear algebra in which two vectors are considered orthogonal if their dot product equals zero. This definition implies that the vectors are at right angles relative to each other in a Euclidean space. Understanding orthogonality is essential for determining the geometrical relationships between vectors, which is particularly important when constructing bases that simplify many problems in geometry and applied mathematics.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Its computation, which involves multiplying corresponding entries and summing the results, is central to determining orthogonality between vectors, as well as measuring projections and angles. Mastery of the dot product is key to exploring relationships in Euclidean space and in higher-dimensional analogues.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be represented as a linear combination of the others. This concept ensures that each vector contributes uniquely to the span of the set, which is essential for forming a basis in a vector space. In contrast, if a set of vectors is linearly dependent, then one or more vectors can be omitted without changing the span, indicating redundancy in the set.

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