Question

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3,0\rangle, \mathbf{v}=\langle 1,1,1\rangle\)

Short answer

Question: Express the vector **u** = <4, 3, 0> as the sum of a vector **p** parallel to **v** = <1, 1, 1> and a vector **n** orthogonal to **v**. Answer: **u** = **p** + **n** = <(7/3), (7/3), (7/3)> + <(5/3), (2/3), (-7/3)> = <4, 3, 0>

Step-by-step solution

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) (vector \(\mathbf{p}\))

To project vector \(\mathbf{u}\) onto the direction of the vector \(\mathbf{v}\), we can use the following formula: $$\mathbf{p} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}$$ First, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\): $$\mathbf{u} \cdot \mathbf{v} = \langle 4,3,0\rangle \cdot \langle 1,1,1\rangle = 4 + 3 + 0 = 7$$ Now, find the norm squared (\(\|\mathbf{v}\|^2\)) of vector \(\mathbf{v}\): $$\|\mathbf{v}\|^2 = 1^2 + 1^2 + 1^2 = 3$$ Now we have the necessary components to calculate the projection vector \(\mathbf{p}\): $$\mathbf{p} = \frac{7}{3} \langle 1,1,1\rangle = \left\langle \frac{7}{3}, \frac{7}{3}, \frac{7}{3}\right\rangle$$

Calculate the orthogonal component (vector \(\mathbf{n}\))

To find the orthogonal component \(\mathbf{n}\), subtract the projection vector \(\mathbf{p}\) from the original vector \(\mathbf{u}\), like this: $$\mathbf{n} = \mathbf{u} - \mathbf{p} = \langle 4,3,0\rangle - \left\langle \frac{7}{3}, \frac{7}{3}, \frac{7}{3}\right\rangle = \left\langle \frac{5}{3}, \frac{2}{3}, -\frac{7}{3}\right\rangle$$ Finally, we can express \(\mathbf{u}\) as the sum of the parallel and orthogonal components, as follows: $$\mathbf{u} = \mathbf{p} + \mathbf{n} = \left\langle \frac{7}{3}, \frac{7}{3}, \frac{7}{3}\right\rangle + \left\langle \frac{5}{3}, \frac{2}{3}, -\frac{7}{3}\right\rangle = \langle 4,3,0\rangle$$

Key concepts

Vector Projection

The concept of vector projection helps us find the closest point on a line (defined by a vector) to another point. This is particularly useful in many fields, such as physics and computer graphics, where understanding the relationship between two vectors is crucial.
To project a vector \( \mathbf{u} \) onto another vector \( \mathbf{v} \), we use the projection formula:

• \( \mathbf{p} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v} \)

This formula uses the dot product of the two vectors, \( \mathbf{u} \) and \( \mathbf{v} \), as well as the magnitude of \( \mathbf{v} \) squared. The result is a vector \( \mathbf{p} \) that lies on the line of \( \mathbf{v} \) and is referred to as the projection of \( \mathbf{u} \) onto \( \mathbf{v} \).
The importance of projecting one vector onto another can be seen when decomposing vectors into parts – one that aligns with a given vector and one that does not. This facilitates analysis and problem-solving in various applications. By calculating \( \mathbf{p} \), we determine which part of \( \mathbf{u} \) runs along \( \mathbf{v} \).

Orthogonal Vectors

Orthogonal vectors are vectors that meet at right angles (90 degrees) to each other. In simple terms, if the dot product of two vectors is zero, they are orthogonal.
Orthogonality can be useful when analyzing vector decomposition. When decomposing a vector \( \mathbf{u} \) into the sum of another vector \( \mathbf{v} \), and a vector \( \mathbf{n} \) which is orthogonal to \( \mathbf{v} \), we find:

• \( \mathbf{u} = \mathbf{p} + \mathbf{n} \)

Where \( \mathbf{p} \) is the projection of \( \mathbf{u} \) onto \( \mathbf{v} \) and \( \mathbf{n} \) is the remaining component orthogonal to \( \mathbf{v} \).
This property is beneficial because it allows for a clear separation of a vector into a component that parallels another direction, and a component that differs entirely from it. This is crucial for applications where such decomposition helps simplify problem analysis, such as signal processing and physics calculations. It assures that changes in one direction do not influence directions orthogonal to it.

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors which results in a scalar (a single number) rather than another vector.
For two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), the dot product is calculated as follows:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

The dot product provides key information about the angle between the vectors:

• If the dot product is positive, the vectors point in more generally the same direction.
• If negative, they point in generally opposite directions.
• If zero, the vectors are orthogonal.

In the context of vector projection, the dot product helps determine the scalar part of the projection formula:\( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \). This scalar part reflects how much of vector \( \mathbf{u} \) 'points' in the same direction as \( \mathbf{v} \). Understanding and utilizing the dot product is essential for decomposing vectors, analyzing directions, and gaining insight into spatial relationships between vectors.