Question

Decomposing vectors 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 \(\mathbf{v}\). $$\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\langle 1,1\rangle$$

Short answer

Question: Given the vectors \(\mathbf{u} = \langle 4, 3\rangle\) and \(\mathbf{v} = \langle 1, 1\rangle\), express \(\mathbf{u}\) as the sum of two vectors \(\mathbf{p}\) and \(\mathbf{n}\), where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\).

Step-by-step solution

Calculate the projection coefficient of \(\mathbf{u}\) onto \(\mathbf{v}\)

First, we need to calculate the projection coefficient of \(\mathbf{u}\) onto \(\mathbf{v}\). We can do this using the dot product formula: $$c = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2}$$ We are given \(\mathbf{u} = \langle 4, 3\rangle\) and \(\mathbf{v} = \langle 1, 1\rangle\). Plugging these values into the formula, we can find the projection coefficient \(c\).

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

Using the projection coefficient \(c\) that we just found, we can calculate the projection vector \(\mathbf{p}\) of \(\mathbf{u}\) onto \(\mathbf{v}\). The projection vector \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and is given by: $$\mathbf{p} = c\mathbf{v}$$

Calculate the orthogonal component \(\mathbf{n}\) of \(\mathbf{u}\) with respect to \(\mathbf{v}\)

Finally, we need to determine the component of \(\mathbf{u}\) that is orthogonal to \(\mathbf{v}\). We can do this by subtracting the projection vector \(\mathbf{p}\) from the original vector \(\mathbf{u}\): $$\mathbf{n} = \mathbf{u} - \mathbf{p}$$ By calculating the orthogonal component \(\mathbf{n}\), we will have expressed \(\mathbf{u}\) as the sum \(\mathbf{u} = \mathbf{p} + \mathbf{n}\), where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\).

Key concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It takes two vectors and returns a scalar. This scalar is a measure of how much one vector extends in the direction of another.
To calculate the dot product of two vectors, say \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), we use the formula:

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

Understanding the dot product is crucial for finding the projection of vectors, as shown in our exercise. In the context of the exercise, the dot product helps find the projection coefficient, indicating how much of \(\mathbf{u}\) aligns along \(\mathbf{v}\).

Projection of Vectors

Projection essentially involves focusing a vector onto the direction of another vector. To put it simply, the projection of vector \(\mathbf{u}\) onto \(\mathbf{v}\) means determining what portion of \(\mathbf{u}\) lies in the direction of \(\mathbf{v}\).
The formula for the projection \(\mathbf{p}\) of \(\mathbf{u}\) on \(\mathbf{v}\) is derived using the dot product:

• Calculate the projection coefficient: \(c = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2}\)
• Determine the projection vector: \(\mathbf{p} = c \mathbf{v}\)

Through this method, we find a vector parallel to \(\mathbf{v}\), allowing us to express \(\mathbf{u}\) as part of this line.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle. In mathematical terms, two vectors are orthogonal if their dot product is zero.
This property makes orthogonal vectors unique, especially when decomposing a vector into components.
In the original exercise, after projecting \(\mathbf{u}\) onto \(\mathbf{v}\), the orthogonal part \(\mathbf{n}\) is found by:

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

Here, vector \(\mathbf{n}\) is perpendicular to \(\mathbf{v}\), showing that \(\mathbf{u}\) is composed of a part along \(\mathbf{v}\) and a part orthogonal to \(\mathbf{v}\). This decomposition simplifies complex vector relations into basic perpendicular components.