91Ó°ÊÓ

For the following exercises, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. a. Find the vector projection \(\mathrm{w}=\mathrm{proj}_{\mathrm{u}} \mathrm{v}\) of vector \(\mathrm{v}\) onto vector u. Express your answer in component form. b. Find the scalar projection comp_ \(\mathrm{v}\) of vector \(\mathbf{v}\) onto vector \(\mathbf{u} .\) Consider the vectors \(\mathbf{u}=4 \mathbf{i}-3 \mathbf{j}\) and \(\mathbf{v}=3 \mathbf{i}+2 \mathbf{j}\) a. Find the component form of vector \(\mathbf{w}=\operatorname{proj}_{\mathbf{u}} \mathbf{v}\) that represents the projection of v onto \(\mathbf{u} .\) b. Write the decomposition \(\mathbf{v}=\mathbf{w}+\mathbf{q}\) of vector \(\mathbf{v}\) into the orthogonal components \(\mathbf{w}\) and \(\mathbf{q}\) , where \(\mathbf{w}\) is the projection of \(\mathbf{v}\) onto \(\mathbf{u}\) and \(\mathbf{q}\) is a vector orthogonal to the direction of \(\mathbf{u} .\)

Step by step solution

01

Understanding the Problem

We are given two vectors \( \mathbf{u} = 4\mathbf{i} - 3\mathbf{j} \) and \( \mathbf{v} = 3\mathbf{i} + 2\mathbf{j} \). We need to find the projection of \( \mathbf{v} \) onto \( \mathbf{u} \), express it in component form, as well as decompose \( \mathbf{v} \) into two components: one along \( \mathbf{u} \) and another orthogonal to \( \mathbf{u} \).

02

Finding the Dot Product

First, calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \). This will be used in finding the projection.\[ \mathbf{u} \cdot \mathbf{v} = (4)(3) + (-3)(2) = 12 - 6 = 6 \]

03

Calculating Magnitude of \( \mathbf{u} \) for Projection

Next, calculate the magnitude of \( \mathbf{u} \) squared, \( \| \mathbf{u} \|^2 \), which is the sum of the squares of its components.\[ \| \mathbf{u} \|^2 = 4^2 + (-3)^2 = 16 + 9 = 25 \]

04

Finding Vector Projection

The vector projection of \( \mathbf{v} \) onto \( \mathbf{u} \) is calculated using:\[ \operatorname{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|^2} \mathbf{u} \]Substituting the values:\[ \operatorname{proj}_{\mathbf{u}} \mathbf{v} = \frac{6}{25} (4\mathbf{i} - 3\mathbf{j}) = \frac{24}{25} \mathbf{i} - \frac{18}{25} \mathbf{j} \]

05

Calculating Scalar Projection

The scalar projection, also known as the component \( \mathrm{comp}_{\mathbf{u}} \mathbf{v} \), is calculated by dividing the dot product by the magnitude of \( \mathbf{u} \).\[ \mathrm{comp}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|} = \frac{6}{\sqrt{25}} = \frac{6}{5} \]

06

Decomposing \( \mathbf{v} \) into \( \mathbf{w} + \mathbf{q} \)

Now, we need to decompose \( \mathbf{v} \) such that \( \mathbf{v} = \mathbf{w} + \mathbf{q} \), where \( \mathbf{w} = \operatorname{proj}_{\mathbf{u}} \mathbf{v} \) and \( \mathbf{q} \) is orthogonal to \( \mathbf{u} \).Using the vector projection for \( \mathbf{w} \):\( \mathbf{w} = \frac{24}{25} \mathbf{i} - \frac{18}{25} \mathbf{j} \)Then, \( \mathbf{q} = \mathbf{v} - \mathbf{w} \):\( \mathbf{q} = (3\mathbf{i} + 2\mathbf{j}) - \left( \frac{24}{25}\mathbf{i} - \frac{18}{25}\mathbf{j} \right) = \left( 3 - \frac{24}{25} \right) \mathbf{i} + \left( 2 + \frac{18}{25} \right) \mathbf{j} \)Calculating these:\( \mathbf{q} = \left( \frac{75}{25} - \frac{24}{25} \right) \mathbf{i} + \left( \frac{50}{25} + \frac{18}{25} \right) \mathbf{j} = \frac{51}{25} \mathbf{i} + \frac{68}{25} \mathbf{j} \)

07

Final Expression

Thus, the decomposition is:\( \mathbf{v} = \left( \frac{24}{25} \mathbf{i} - \frac{18}{25} \mathbf{j} \right) + \left( \frac{51}{25} \mathbf{i} + \frac{68}{25} \mathbf{j} \right) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dot Product

The dot product is a way to multiply two vectors and get a scalar, or simply a number, as a result. It is an important tool in finding projections and angles between vectors.
To compute the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \), you simply multiply corresponding components and sum the results:

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

In this exercise, \( \mathbf{u} = 4\mathbf{i} - 3\mathbf{j} \) and \( \mathbf{v} = 3\mathbf{i} + 2\mathbf{j} \).
By calculating:

• \( \mathbf{u} \cdot \mathbf{v} = (4)(3) + (-3)(2) = 6 \)

This dot product is essential for finding the vector projection of one vector onto another.

Magnitude of a Vector

The magnitude of a vector, often represented as \( \| \mathbf{u} \| \), is like the vector's length. It is calculated using the Pythagorean theorem.

• For any vector \( \mathbf{u} = (u_1, u_2) \), its magnitude is \( \| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2} \)

In calculating projections, we use the square of the magnitude, \( \| \mathbf{u} \|^2 \), which simplifies the math in projection formulas.
For \( \mathbf{u} = 4\mathbf{i} - 3\mathbf{j} \), the magnitude squared is:

• \( \| \mathbf{u} \|^2 = 4^2 + (-3)^2 = 25 \)

This value helps determine how much of vector \( \mathbf{v} \) lines up with vector \( \mathbf{u} \).

Orthogonal Decomposition

Orthogonal decomposition refers to breaking down a vector into two perpendicular parts. One part, \( \mathbf{w} \), aligns with another vector, and the second part, \( \mathbf{q} \), is orthogonal to it.

• For \( \mathbf{v} \), \( \mathbf{v} = \mathbf{w} + \mathbf{q} \)

Orthogonality means that the dot product of \( \mathbf{w} \) and \( \mathbf{q} \) is zero, indicating they form a right angle. In our exercise:

• \( \mathbf{w} \) is the projection of \( \mathbf{v} \) onto \( \mathbf{u} \)
• \( \mathbf{q} \) is the remainder, represented by \( \mathbf{q} = \mathbf{v} - \mathbf{w} \)

This confirms that each component vector is perpendicular to the other, illustrating the concept of orthogonal components.

Scalar Projection

The scalar projection of a vector \( \mathbf{v} \) onto another vector \( \mathbf{u} \) captures how much of \( \mathbf{v} \) lies in the direction of \( \mathbf{u} \). It is a single number providing the magnitude without direction.
The formula used is:

• \( \mathrm{comp}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf{u} \|} \)

For the vectors in the exercise:

• \( \mathrm{comp}_{\mathbf{u}} \mathbf{v} = \frac{6}{5} \)

This scalar gives a straightforward measure of how much \( \mathbf{v} \) travels along the line defined by \( \mathbf{u} \). It is especially useful when considering movements or forces in a particular direction.