The figure shows vector a⃗ of magnitude 2 m, and vector b⃗ ofmagnitude 4 m. Segments OP and SU are parallel; segments OS and PU areparallel.Evaluate the unit vector parallel to the direction that goes from O to U.Justify ALL your steps. Be very explicit in your answer. ## Vector Addition and Parallelogram Law### Diagram ExplanationThe provided diagram represents a classical example of vector addition using the parallelogram law. Let's break down the elements in the diagram:1. **Axes**: - X-axis: Horizontal axis labeled as \( X \), with a unit vector \( \hat{i} \) pointing to the right. - Y-axis: Vertical axis labeled as \( Y \), with a unit vector \( \hat{j} \) pointing upwards.2. **Vectors**: - **Vector \( \vec{a} \)** is represented by the blue arrow \( OP \) starting at point \( O \) (the origin) and ending at point \( P \). This vector makes an angle of \( 30^\circ \) with the Y-axis. - **Vector \( \vec{b} \)** is represented by the blue arrow \( OS \) starting at point \( O \) and ending at point \( S \). This vector makes an angle of \( 60^\circ \) with the Y-axis.3. **Construction**: - From point \( P \), draw a line parallel to \( \vec{b} \), ending at point \( U \). - From point \( S \), draw a line parallel to \( \vec{a} \), also ending at point \( U \).4. **Resultant Vector**: - The diagonal \( OU \) of the parallelogram \( OPSU \) represents the resultant vector of \( \vec{a} \) and \( \vec{b} \).### Mathematical RepresentationThe parallelogram law of vector addition states that if two vectors \( \vec{a} \) and \( \vec{b} \) are represented as adjacent sides of a parallelogram, then the resultant vector \( \vec{R} \) is represented by the diagonal of the parallelogram starting from the same point. \[ \vec{R} = \vec{a} + \vec{b} \]### Angular Relationships- The angle between the positive Y-axis and \( \vec{a} \) is \( 30^\circ \).- The angle between the positive Y-axis and \( \vec{b} \) is \( 60^\circ \).This diagram visually explains how vectors can be added together using the parallelogram law,

This is the question from vector analysis.

The figure shows vector a⃗ of magnitude 2 m, and vector b⃗ of magnitude 4 m. Segments OP and SU are parallel; segments OS and PU are parallel. Evaluate the unit vector parallel to the direction that goes from O to U. Justify ALL your steps. Be very explicit in your answer.

The figure shows vector a⃗ of magnitude 2 m, and vector b⃗ of magnitude 4 m. Segments OP and SU are parallel; segments OS and PU are parallel. Evaluate the unit vector parallel to the direction that goes from O to U. Justify ALL your steps. Be very explicit in your answer.

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

The figure shows vector a⃗ of magnitude 2 m, and vector b⃗ of
magnitude 4 m. Segments OP and SU are parallel; segments OS and PU are
parallel.
Evaluate the unit vector parallel to the direction that goes from O to U.
Justify ALL your steps. Be very explicit in your answer.

## Vector Addition and Parallelogram Law

### Diagram Explanation
The provided diagram represents a classical example of vector addition using the parallelogram law. Let's break down the elements in the diagram:

1. **Axes**:
- X-axis: Horizontal axis labeled as \( X \), with a unit vector \( \hat{i} \) pointing to the right.
- Y-axis: Vertical axis labeled as \( Y \), with a unit vector \( \hat{j} \) pointing upwards.

2. **Vectors**:
- **Vector \( \vec{a} \)** is represented by the blue arrow \( OP \) starting at point \( O \) (the origin) and ending at point \( P \). This vector makes an angle of \( 30^\circ \) with the Y-axis.
- **Vector \( \vec{b} \)** is represented by the blue arrow \( OS \) starting at point \( O \) and ending at point \( S \). This vector makes an angle of \( 60^\circ \) with the Y-axis.

3. **Construction**:
- From point \( P \), draw a line parallel to \( \vec{b} \), ending at point \( U \).
- From point \( S \), draw a line parallel to \( \vec{a} \), also ending at point \( U \).

4. **Resultant Vector**:
- The diagonal \( OU \) of the parallelogram \( OPSU \) represents the resultant vector of \( \vec{a} \) and \( \vec{b} \).

### Mathematical Representation
The parallelogram law of vector addition states that if two vectors \( \vec{a} \) and \( \vec{b} \) are represented as adjacent sides of a parallelogram, then the resultant vector \( \vec{R} \) is represented by the diagonal of the parallelogram starting from the same point.

\[ \vec{R} = \vec{a} + \vec{b} \]

### Angular Relationships
- The angle between the positive Y-axis and \( \vec{a} \) is \( 30^\circ \).
- The angle between the positive Y-axis and \( \vec{b} \) is \( 60^\circ \).

This diagram visually explains how vectors can be added together using the parallelogram law,

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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