Question

Express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of the components of the vectors.

Short answer

Answer: The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be found using the formula: \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2} + \cdots + u_{n}v_{n}\) where \(u_i\) and \(v_i\) represent the components of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), respectively.

Step-by-step solution

Understand what a dot product is

The dot product (also known as the scalar product or inner product) is an operation that takes two vectors and returns a scalar value. This value is a measure of the similarity between the two vectors. If the dot product of two vectors is equal to 0, it means that the vectors are orthogonal (perpendicular) to each other.

Introduce the vectors

Let \(\mathbf{u}\) and \(\mathbf{v}\) be two arbitrary vectors in \(\mathbb{R}^n\) such that: \(\mathbf{u} = \begin{pmatrix} u_{1}\\ u_{2}\\ \vdots\\ u_{n} \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} v_{1}\\ v_{2}\\ \vdots\\ v_{n} \end{pmatrix}\)

Calculate the dot product

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is defined as: \(\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_{i}v_{i}\) where \(u_i\) and \(v_i\) represent the components of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), respectively.

Write the final expression

Now, we can express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of the components of the vectors: \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2} + \cdots + u_{n}v_{n}\)