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The dot product is a crucial operation in understanding orthogonality in vector spaces. It's a way to multiply two vectors to get a scalar. To compute the dot product, align the corresponding components of two vectors, multiply each pair, and then sum all these products. For vectors \(\mathbf{a} = [a_1, a_2, a_3]\) and \(\mathbf{b} = [b_1, b_2, b_3]\), the dot product is calculated as \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\).
In the context of checking for an orthogonal basis, a set of vectors in \(\mathbb{R}^n\) is orthogonal when the dot product between every pair of distinct vectors in the set equals zero. This property is essential, as it simplifies many calculations, including vector decompositions. In this exercise, verifying that vectors \(\mathbf{u}_1, \mathbf{u}_2,\) and \(\mathbf{u}_3\) are orthogonal was the first step in ensuring they form an orthogonal basis.
Orthogonal vectors not only simplify the process of breaking down other vectors (decomposition) but also ensure the absence of redundancy within the set, which is beneficial for constructing an efficient and concise basis for a vector space.

A linear combination involves expressing a vector as a sum of scalar multiples of other vectors. It's a method used to describe vectors within a space in terms' basis vectors. For instance, when you have an orthogonal basis, an arbitrary vector \(\mathbf{x}\) can be expressed as a linear combination of the basis vectors, such that \(\mathbf{x} = c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + c_3\mathbf{u}_3\).
To find the coefficients (\(c_1, c_2, c_3\)), you use the dot product. In orthogonal bases, computing these coefficients is straightforward:

• Take the dot product of the vector \(\mathbf{x}\) with each basis vector \(\mathbf{u}_i\).
• Divide by the dot product of \(\mathbf{u}_i\) with itself, \(\mathbf{u}_i \cdot \mathbf{u}_i\).

Thus, each coefficient corresponds to how much of each basis vector is needed to "build" the vector \(\mathbf{x}\). Orthogonal bases simplify the calculation of these coefficients, making the representation clear and arithmetic manageable.

Vector decomposition is a technique that involves breaking a vector down into a sum of vectors, often relative to a particular basis set. This is exceedingly useful for simplifying complex vector operations and solving various mathematical problems.
In the exercise, vector decomposition is carried out with an orthogonal basis, making it especially straightforward. With the vector \(\mathbf{x}\) given and an orthogonal basis \(\{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\}\), we decompose \(\mathbf{x}\) using a formula that involves the dot product:

• Each component of \(\mathbf{x}\) relative to the basis vector \(\mathbf{u}_i\) is determined by \(c_i = \frac{\mathbf{x} \cdot \mathbf{u}_i}{\mathbf{u}_i \cdot \mathbf{u}_i}\).

In practice, the decomposition allows us to reassemble any target vector as a weighted sum of simpler, orthogonal vectors. This method has applications in everything from physics to computer graphics, anytime a complex vector needs to be analyzed or transformed against a given basis.