### Orthogonal Projection Problem**Problem Statement:**Compute the orthogonal projection of $\begin{bmatrix} -1 \\ 4 \end{bmatrix}$ onto the line through $\begin{bmatrix} -3 \\ 5 \end{bmatrix}$ and the origin.**Solution:**The orthogonal projection is $\begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}$.Enter your answer in the edit fields and then click "Check Answer."**Note:**For computing the orthogonal projection of a vector $\mathbf{a}$ onto another vector $\mathbf{b}$, use the formula:\[\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}\]where $\mathbf{a} \cdot \mathbf{b}$ is the dot product of vectors $\mathbf{a}$ and $\mathbf{b}$. ### Instructions:1. Calculate the dot product of the vectors.2. Find the magnitude squared of vector $\mathbf{b}$.3. Use the projection formula to find the result.4. Fill in the boxes provided with the correct values.**Diagram and Explanation:**(In this case, there are no diagrams provided. An explanation of steps and formulae is included above.)After computing, enter your answers and click "Check Answer" to verify.### Example Calculation:Given:$\mathbf{a} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}, \mathbf{b} = \begin{bmatrix} -3 \\ 5 \end{bmatrix}$1. Compute the dot product $\mathbf{a} \cdot \mathbf{b}$:\[\mathbf{a} \cdot \mathbf{b} = (-1)(-3) + (4)(5) = 3 + 20 = 23\]2. Compute $\mathbf{b} \cdot \mathbf{b}$:\[\mathbf{b} \cdot \mathbf{b} = (-3)^2 + 5^2 = 9 + 25 = 34\]3. Use the projection

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Answered: -1 onto the line through 4 -3 and the…

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-1 onto the line through 4 -3 and the origin. Compute the orthogonal projection of The orthogonal projection is

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Orthogonal Projection Problem

**Problem Statement:**
Compute the orthogonal projection of $\begin{bmatrix} -1 \\ 4 \end{bmatrix}$ onto the line through $\begin{bmatrix} -3 \\ 5 \end{bmatrix}$ and the origin.

**Solution:**

The orthogonal projection is $\begin{bmatrix} \boxed{} \\ \boxed{} \end{bmatrix}$.

Enter your answer in the edit fields and then click "Check Answer."

**Note:**
For computing the orthogonal projection of a vector $\mathbf{a}$ onto another vector $\mathbf{b}$, use the formula:

\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}
\]

where $\mathbf{a} \cdot \mathbf{b}$ is the dot product of vectors $\mathbf{a}$ and $\mathbf{b}$.

### Instructions:
1. Calculate the dot product of the vectors.
2. Find the magnitude squared of vector $\mathbf{b}$.
3. Use the projection formula to find the result.
4. Fill in the boxes provided with the correct values.

**Diagram and Explanation:**
(In this case, there are no diagrams provided. An explanation of steps and formulae is included above.)

After computing, enter your answers and click "Check Answer" to verify.

### Example Calculation:
Given:
$\mathbf{a} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}, \mathbf{b} = \begin{bmatrix} -3 \\ 5 \end{bmatrix}$

1. Compute the dot product $\mathbf{a} \cdot \mathbf{b}$:
\[
\mathbf{a} \cdot \mathbf{b} = (-1)(-3) + (4)(5) = 3 + 20 = 23
\]

2. Compute $\mathbf{b} \cdot \mathbf{b}$:
\[
\mathbf{b} \cdot \mathbf{b} = (-3)^2 + 5^2 = 9 + 25 = 34
\]

3. Use the projection

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