Question

Properties of dot products 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 \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. Distributive properties a. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+2 \mathbf{u} \cdot \mathbf{v}+|\mathbf{v}|^{2}\) b. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+|\mathbf{v}|^{2}\) if \(\mathbf{u}\) is orthogonal to \(\mathbf{v}\) c. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}-\mathbf{v})=|\mathbf{u}|^{2}-|\mathbf{v}|^{2}\)

Short answer

Question: Prove the following vector properties involving the dot product, distribution, and orthogonality: a) Show that $(\mathbf{u}+\mathbf{v})\cdot(\mathbf{u}+\mathbf{v}) = |\mathbf{u}|^2+2\mathbf{u}\cdot\mathbf{v}+|\mathbf{v}|^2$. b) If $\mathbf{u}$ is orthogonal to $\mathbf{v}$, show that $(\mathbf{u}+\mathbf{v})\cdot(\mathbf{u}+\mathbf{v}) = |\mathbf{u}|^2 + |\mathbf{v}|^2$. c) Show that $(\mathbf{u}+\mathbf{v})\cdot(\mathbf{u}-\mathbf{v}) = |\mathbf{u}|^2 - |\mathbf{v}|^2$.

Step-by-step solution

Part a: Distributive Property Proof 1

First, compute the dot product of \((\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}+\mathbf{v})\): $$(\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}+\mathbf{v}) = (u_1 + v_1)(u_1 + v_1) + (u_2 + v_2)(u_2 + v_2) + (u_3 + v_3)(u_3 + v_3)$$ Now, we can expand and simplify the expression: $$u_1^2 + 2u_1v_1 + v_1^2 + u_2^2 + 2u_2v_2 + v_2^2 + u_3^2 + 2u_3v_3 + v_3^2$$ Finally, we can rewrite the expression to match the desired result: $$|\mathbf{u}|^{2}+2 \mathbf{u} \cdot\mathbf{v}+|\mathbf{v}|^{2}$$

Part b: Distributive Property Proof 2

For this case, we are given that \(\mathbf{u}\) is orthogonal to \(\mathbf{v}\), which means their dot product is zero: \(\mathbf{u} \cdot \mathbf{v} = 0\). Using the result from part a, we have: $$(\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}+\mathbf{v}) = |\mathbf{u}|^{2}+2 \mathbf{u} \cdot\mathbf{v}+|\mathbf{v}|^{2}$$ As \(\mathbf{u} \cdot \mathbf{v} = 0\), the equation simplifies to: $$|\mathbf{u}|^{2}+|\mathbf{v}|^{2}$$

Part c: Distributive Property Proof 3

First, compute the dot product of \((\mathbf{u}+\mathbf{v})\cdot(\mathbf{u}-\mathbf{v})\): $$(\mathbf{u}+\mathbf{v})\cdot(\mathbf{u}-\mathbf{v}) = (u_1 + v_1)(u_1 - v_1) + (u_2 + v_2)(u_2 - v_2) + (u_3 + v_3)(u_3 - v_3)$$ Now, we can expand and simplify the expression: $$u_1^2 - u_1v_1 + u_1v_1 - v_1^2 + u_2^2 - u_2v_2 + u_2v_2 - v_2^2 + u_3^2 - u_3v_3 + u_3v_3 - v_3^2$$ Finally, we can rewrite the expression to match the desired result: $$|\mathbf{u}|^{2}-|\mathbf{v}|^{2}$$

Key concepts

Vector Calculus

Vector calculus is a branch of mathematics that involves differentiation and integration of vector fields. A vector has both magnitude and direction, which makes vector calculus essential for dealing with multidimensional spaces and situations like physics and engineering. In vector calculus, operations like dot product and cross product are used often. These operations help to find relationships between vectors, analyze the space they occupy, and provide insights into fields like electromagnetism and fluid dynamics.
### Understanding Vectors - **Magnitude**: The length of a vector, denoted as \(|\mathbf{u}|\), calculated using the Pythagorean theorem - **Direction**: Determines where the vector is pointing - **Dot Product (Scalar Product)**: A fundamental operation where two vectors are multiplied. It provides a scalar result. The formula for the dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is: \[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \] This can show alignment or the projection of one vector onto another.
These concepts are elemental to understanding how vectors interact within different fields, providing mathematical tools to solve real-world problems.

Orthogonal Vectors

Orthogonality in vectors describes a situation where two vectors are at right angles (90 degrees) to each other. In mathematical terms, when two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, their dot product equals zero: \(\mathbf{u} \cdot \mathbf{v} = 0\). This is because the component of one vector along the other is zero.
### Characteristics of Orthogonal Vectors - **Right Angle**: Represented graphically, these vectors would create a 90-degree angle at the origin when placed tail to tail. - **Zero Projection**: Neither vector projects onto the other, which geometrically means they hold no influence in each other's direction. - **Use in Applications**: Orthogonal vectors are used in many fields. For instance, in computer graphics to ensure coordinate axes are perpendicular, in image processing to differentiate between features, or in statistics to ensure that different data dimensions remain independent.
Understanding and utilizing orthogonal vectors is important in both practical and theoretical aspects as they simplify calculations and provide clear insights into vector spaces.

Distributive Property

The distributive property is a fundamental rule in both arithmetic and algebra which extends to operations on vectors. It states that for any vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), the dot product satisfies: \[ \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \] This property makes it possible to simplify expressions and proofs involving vectors by unravelling the operations systematically.
**In Vector Calculus**: The distributive property can be applied when expanding expressions like \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v})\). It helps in proving equations and understanding the relationship between different vector operations.
### Applications- **Algebraic Simplification**: Essential in simplifying vector expressions by breaking them down into smaller parts that are easier to compute. - **Vector Projection and Analysis**: Used in analyzing forces and motions in physics, where vectors often represent quantities like force and velocity.
Through these aspects, the distributive property connects algebra with geometry, giving insight into how vectors add and multiply in space.