Problem 27 - Precalculus

Question

Find the vector, given its magnitude and direction angle. $$|\mathbf{u}|=16, \theta=100^{\circ}$$

Step-by-Step Solution

Answer

The vector is approximately \(\langle -2.79, 15.76 \rangle\).

Step 1: Understanding the Problem

We need to find the components of a vector given its magnitude, \(|\mathbf{u}|=16\), and its direction angle \(\theta=100^{\circ}\). This involves converting polar coordinates (magnitude and angle) to Cartesian coordinates (vector components).

Step 2: Formula for the Components

The vector \(\mathbf{u}\) can be represented in component form as \(\mathbf{u} = \langle u_x, u_y \rangle\) where \(u_x = |\mathbf{u}| \cos(\theta)\) and \(u_y = |\mathbf{u}| \sin(\theta)\).

Step 3: Calculating the X Component

Using the formula for the x-component: \(u_x = |\mathbf{u}| \cos(\theta) = 16 \cos(100^{\circ})\). Use a calculator to find \(\cos(100^{\circ})\) and multiply by 16.

Step 4: Calculating the Y Component

Using the formula for the y-component: \(u_y = |\mathbf{u}| \sin(\theta) = 16 \sin(100^{\circ})\). Use a calculator to find \(\sin(100^{\circ})\) and multiply by 16.

Step 5: Compute and Combine Components

Find \(u_x ≈ 16 \cos(100^{\circ}) ≈ -2.79\) and \(u_y ≈ 16 \sin(100^{\circ}) ≈ 15.76\). Thus, the vector \(\mathbf{u} ≈ \langle -2.79, 15.76 \rangle\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing a point or a vector in a plane using two values: the radial distance and the angle from a reference direction. These two values are:

• Magnitude (or radius): This is the length of the vector measured from the origin to the point in question. In this exercise, the magnitude is 16.
• Direction (or angle): This is measured in degrees or radians and specifies the angle from a reference direction, typically the positive x-axis. For this problem, the direction angle is 100°.

The advantage of polar coordinates is that they can simplify the understanding of problems involving angles and distances. They are particularly useful in situations where the relationship with angles needs to be considered prominently.

Cartesian Coordinates

Cartesian coordinates express a point or vector using two perpendicular axes, usually labeled as x and y. This system is based on a grid defined by these axes, allowing you to precisely position any point with two numeric coordinates:

• X-coordinate (horizontal component): Specifies the distance along the x-axis.
• Y-coordinate (vertical component): Specifies the distance along the y-axis.

To convert polar coordinates to Cartesian coordinates, you use the angle and magnitude given in polar form. The conversion for our problem is done using:

• For x-component: \( u_x = |\mathbf{u}| \cos(\theta) \)
• For y-component: \( u_y = |\mathbf{u}| \sin(\theta) \)

This process transforms our initial polar coordinates (16, 100°) into the more easily manageable Cartesian form, making it easier to perform vector mathematics like addition, subtraction, and scaling.

Magnitude and Direction

Magnitude and direction are the core characteristics of any vector. Understanding these helps in visualizing how vectors influence each other in space:

• Magnitude: Represents the length or size of the vector. It is a non-negative number, indicating how far it reaches in its designated direction. In our exercise, the vector has a magnitude of 16.
• Direction: Specifies where the vector is pointing relative to a reference direction. In our case, the direction is set by an angle of 100° from the positive x-axis.

When you know a vector's magnitude and direction, you can perform several operations:

• Convert it from polar to Cartesian coordinates to find its components.
• Understand the resultant force or movement when vectors are combined.
• Predict the motion of objects or understand fields in physics.

This exercise revolves around how we can translate the simple ideas of magnitude and direction into practices that allow precise modeling in mathematical and real-world contexts.