91Ó°ÊÓ

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It involves two vectors and results in a scalar (a single number) rather than a vector. The dot product is calculated by multiplying corresponding components of the two vectors and then summing these products.

For vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\),the dot product is given by:

• \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \)

This operation is widely used because it can tell you about the relationship between the vectors, such as whether they are perpendicular (in which case the dot product is zero).

In our exercise, we calculated the dot products \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{u} \cdot \mathbf{w}\), which were used to verify the vector identity.

Vector addition is the process of combining two or more vectors to form a new vector. This is done by adding corresponding components of the vectors.

For any two vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle \), their sum \(\mathbf{a} + \mathbf{b} \) is computed as:

• First component: \( a_1 + b_1\)
• Second component: \( a_2 + b_2 \)
• Third component: \( a_3 + b_3 \)

This results in a new vector \(\langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle\).

In the context of the exercise, vector addition was used to combine vectors \(\mathbf{v}\) and \(\mathbf{w}\) into a single vector \(\mathbf{v} + \mathbf{w}\). This step laid the foundation for verifying the vector identity.

The vector identity \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\) is an important concept in vector algebra. It exemplifies how distribution works with vectors and dot products—very similar to how distribution works with numbers.

In this identity, the distribution of \(\mathbf{u}\) over the addition of vectors \(\mathbf{v}\) and \(\mathbf{w}\) means that you can calculate the dot product of \(\mathbf{u}\) with \(\mathbf{v}\) and \(\mathbf{w}\) separately, then sum the results.

In our verification problem, this identity was confirmed to hold true, thanks to computation parts:

• The left-hand side \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w})\) was shown to equal 46.
• The right-hand side consisting of the separate dot products \((\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w})\) also summed to 46.

Hence, the identity was affirmed valid for the given vectors, reinforcing the understanding of vector operations.