Question

Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle\), and \(\mathbf{w}=\) \(\left\langle w_{1}, w_{2}, w_{3}\right\rangle\). Let \(c\) be a scalar. Prove the following vector properties. \(\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}\)

Short answer

Question: Prove the commutative property for the dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\): \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\). Answer: Using the definition of the dot product and the commutative property of multiplication, we showed that \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}\) and \(\mathbf{v} \cdot \mathbf{u} = v_{1}u_{1} + v_{2}u_{2} + v_{3}u_{3}\), which are equal expressions. Therefore, \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\), proving the commutative property for the dot product.

Step-by-step solution

Write down the definition of the dot product

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is defined as \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}\).

Calculate \(\mathbf{u} \cdot \mathbf{v}\)

Using the definition of the dot product, we can calculate \(\mathbf{u} \cdot \mathbf{v}\) as \(u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}\).

Calculate \(\mathbf{v} \cdot \mathbf{u}\)

Similarly, we can calculate \(\mathbf{v} \cdot \mathbf{u}\) as \(v_{1}u_{1} + v_{2}u_{2} + v_{3}u_{3}\).

Prove the commutative property

Since multiplication is commutative, we know that \(u_{1}v_{1}=v_{1}u_{1}\), \(u_{2}v_{2}=v_{2}u_{2}\), and \(u_{3}v_{3}=v_{3}u_{3}\). So, the expressions for \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{u}\) are equal: $$u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3} = v_{1}u_{1} + v_{2}u_{2} + v_{3}u_{3}$$ Thus, we have proven the commutative property for the dot product: \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\).

Key concepts

Dot Product

The dot product is an essential concept in vector algebra that involves two vectors. Given two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), their dot product is calculated as the sum of the products of their corresponding components:

• \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \)

This operation results in a scalar, not a vector. The dot product measures the extent to which two vectors point in the same direction.
If the result is positive, the vectors tend to point in the same direction. If it's zero, the vectors are perpendicular.
Understanding the dot product helps explore concepts such as projection and orthogonality, which are fundamental in physics and engineering.

Commutative Property

In mathematics, the commutative property refers to the idea that the order in which two numbers are added or multiplied does not change the result. This holds true for the dot product of vectors as well.
To demonstrate the commutative property for the dot product, consider two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \).
Calculating \( \mathbf{u} \cdot \mathbf{v} \) gives us:

• \( u_1 v_1 + u_2 v_2 + u_3 v_3 \)

Similarly, \( \mathbf{v} \cdot \mathbf{u} \) yields:

• \( v_1 u_1 + v_2 u_2 + v_3 u_3 \)

Since multiplication is commutative\( (a \cdot b = b \cdot a) \), each of these expressions is identical, demonstrating that:

• \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \)

This property makes the dot product symmetrical and convenient for analysis of vector relationships.

Vector Operations

Vector operations encompass various ways to manipulate vectors mathematically. Common operations include:

• Addition and Subtraction: Vectors are added by summing their corresponding components. Subtraction works the same, but instead subtracts components.


• Scalar Multiplication: Each component of the vector is multiplied by a scalar, altering the vector's magnitude but not its direction.


• Dot Product: As we defined earlier, the dot product is a scalar value indicating the directional relationship between two vectors.


• Cross Product: Exclusive to three-dimensional space, the cross product results in a vector perpendicular to the original vectors, representing the area of the parallelogram formed by them.

Understanding vector operations is vital, as they are applied in numerous fields such as physics, engineering, and computer graphics. Such operations allow us to handle spatial objects efficiently, predict motion, and simulate real-world environments. Each operation uniquely addresses specific problems and enhances our ability to work with vector quantities.