### Physics Problems - Vector Magnitudes and Relative Velocity#### Problem 3:In the above problem, what is $|\vec{D}|$, the magnitude of vector $\vec{D} = \vec{A} - \vec{B}$?- (A) 1.0 cm- (B) 1.1 cm- (C) 1.2 cm- (D) 1.4 cm#### Problem 4:A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. What is the velocity relative to an observer on the shore?- (A) 3.2 m/s to the southeast- (B) 5.0 m/s to the southeast- (C) 7.1 m/s to the southeast- (D) 10.0 m/s to the southeast### Explanation:**For Problem 3:**The problem requires determining the magnitude of the resultant vector $\vec{D}$ when vector $\vec{B}$ is subtracted from vector $\vec{A}$. The correct answer can be chosen from options (A) to (D) based on the given numerical values.**For Problem 4:**The problem states that a motorboat moves due east at 5.0 m/s while a river flows south at 5.0 m/s. To find the velocity of the boat relative to an observer on the shore, we need to consider both velocities as perpendicular components of the resultant velocity.- The eastward velocity component: $\vec{v}_\text{boat} = 5.0 \, \text{m/s}$- The southward velocity component: $\vec{v}_\text{river} = 5.0 \, \text{m/s}$Using the Pythagorean theorem:\[ v_{\text{resultant}} = \sqrt{(v_{\text{boat}})^2 + (v_{\text{river}})^2} \]\[ v_{\text{resultant}} = \sqrt{(5.0 \, \text{m/s})^2 + (5.0 \, \text{m/s})^2} \]\[ v_{\text{resultant}} = \sqrt{25 + The image displays a coordinate system with the $x$-axis and $y$-axis. Two vectors are represented in this system:1. Vector $\vec{A}$ is positioned along the positive $x$-axis and has a magnitude of 2.0 cm.2. Vector $\vec{B}$ is oriented at an angle $\alpha = 30^\circ$ relative to the positive $x$-axis. This vector also has a magnitude of 2.0 cm.The image shows that the tail of each vector is positioned at the origin (0,0) of the coordinate system. Vector $\vec{A}$ extends horizontally to the right, while vector $\vec{B}$ extends diagonally upward to the right, forming a 30-degree angle with vector $\vec{A}$.This diagram can be useful for understanding the representation of vectors in a coordinate system, the concept of vector magnitude and direction, and how trigonometric functions can be applied to decompose vectors into their components.

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3. In the above problem, what is, ĮD|, the magnitude of vector Ď = Ä – B ? (A) 1.0 cm (B) 1.1 cm (C) 1.2 cm (D) 1.4 cm

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Physics Problems - Vector Magnitudes and Relative Velocity

#### Problem 3:
In the above problem, what is $|\vec{D}|$, the magnitude of vector $\vec{D} = \vec{A} - \vec{B}$?
- (A) 1.0 cm
- (B) 1.1 cm
- (C) 1.2 cm
- (D) 1.4 cm

#### Problem 4:
A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. What is the velocity relative to an observer on the shore?
- (A) 3.2 m/s to the southeast
- (B) 5.0 m/s to the southeast
- (C) 7.1 m/s to the southeast
- (D) 10.0 m/s to the southeast

### Explanation:

**For Problem 3:**
The problem requires determining the magnitude of the resultant vector $\vec{D}$ when vector $\vec{B}$ is subtracted from vector $\vec{A}$. The correct answer can be chosen from options (A) to (D) based on the given numerical values.

**For Problem 4:**
The problem states that a motorboat moves due east at 5.0 m/s while a river flows south at 5.0 m/s. To find the velocity of the boat relative to an observer on the shore, we need to consider both velocities as perpendicular components of the resultant velocity.
- The eastward velocity component: $\vec{v}_\text{boat} = 5.0 \, \text{m/s}$
- The southward velocity component: $\vec{v}_\text{river} = 5.0 \, \text{m/s}$

Using the Pythagorean theorem:
\[ v_{\text{resultant}} = \sqrt{(v_{\text{boat}})^2 + (v_{\text{river}})^2} \]

\[ v_{\text{resultant}} = \sqrt{(5.0 \, \text{m/s})^2 + (5.0 \, \text{m/s})^2} \]

\[ v_{\text{resultant}} = \sqrt{25 +

The image displays a coordinate system with the $x$-axis and $y$-axis.

Two vectors are represented in this system:
1. Vector $\vec{A}$ is positioned along the positive $x$-axis and has a magnitude of 2.0 cm.
2. Vector $\vec{B}$ is oriented at an angle $\alpha = 30^\circ$ relative to the positive $x$-axis. This vector also has a magnitude of 2.0 cm.

The image shows that the tail of each vector is positioned at the origin (0,0) of the coordinate system. Vector $\vec{A}$ extends horizontally to the right, while vector $\vec{B}$ extends diagonally upward to the right, forming a 30-degree angle with vector $\vec{A}$.

This diagram can be useful for understanding the representation of vectors in a coordinate system, the concept of vector magnitude and direction, and how trigonometric functions can be applied to decompose vectors into their components.

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