Problem 25 - Precalculus

Question

Find the vector, given its magnitude and direction angle. $$|\mathbf{u}|=7, \theta=25^{\circ}$$

Step-by-Step Solution

Answer

The vector is approximately \( \mathbf{u} = \langle 6.342, 2.959 \rangle \).

Step 1: Understanding the Problem

We need to find the vector \( \mathbf{u} \) given its magnitude and direction angle. The magnitude is given as \( |\mathbf{u}| = 7 \) and the direction angle \( \theta = 25^{\circ} \).

Step 2: Set Up the Vector Formula

The component form of a vector \( \mathbf{u} \) given its magnitude \( |\mathbf{u}| \) and direction angle \( \theta \) is \( \mathbf{u} = \langle |\mathbf{u}| \cos(\theta), |\mathbf{u}| \sin(\theta) \rangle \).

Step 3: Substitute the Given Values

Substitute \( |\mathbf{u}| = 7 \) and \( \theta = 25^{\circ} \) into the formula: \( \mathbf{u} = \langle 7 \cos(25^{\circ}), 7 \sin(25^{\circ}) \rangle \).

Step 4: Calculate the Components

Calculate the components using a calculator: \( 7 \cos(25^{\circ}) \approx 6.342 \) and \( 7 \sin(25^{\circ}) \approx 2.959 \).

Step 5: Write the Final Vector

The vector \( \mathbf{u} \) in component form is \( \mathbf{u} \approx \langle 6.342, 2.959 \rangle \).

Key Concepts

Magnitude of a Vector

The magnitude of a vector is essentially its length and is always a non-negative value. It's a measure of how long or short the vector is, regardless of its direction. You can think of magnitude as the 'size' or 'strength'.
For a vector \( \mathbf{u} \) represented in component form \( \langle a, b \rangle \), its magnitude can be calculated using the Pythagorean theorem. The formula is:

• \( |\mathbf{u}| = \sqrt{a^2 + b^2} \)

Given that vectors have a directional element, the magnitude helps separate out just the 'how much' part without considering 'where to.' In our scenario, the vector's magnitude is given as \( |\mathbf{u}| = 7 \). This value plays a crucial role in determining the vector's component form and ultimately its actual representation.

Direction Angle of a Vector

Direction angle is the angle made by the vector with the positive x-axis. In navigation and physics, direction angles help specify the vector's orientation. We measure this angle in degrees or radians, starting from the origin.

• For example, an angle of \( 0^{\circ} \) would indicate a vector pointing directly to the right along the x-axis.
• An angle of \( 90^{\circ} \) points straight up along the y-axis.

The key is to always start at the positive x-axis and move anti-clockwise. In our exercise, the given direction angle is \( \theta = 25^{\circ} \). This tells us the angle at which our vector is pointing from the horizontal line (or x-axis).
Understanding direction angles is fundamental for converting between polar and rectangular coordinates in vector problems.

Component Form of a Vector

The component form of a vector is a way to express the vector using its horizontal and vertical components. It's like describing how far along the x-axis and y-axis the vector moves. For the components, we often use the vector formula derived from trigonometry:

• \( \mathbf{u} = \langle |\mathbf{u}| \cos(\theta), |\mathbf{u}| \sin(\theta) \rangle \)

This formula takes advantage of the fact that a vector can be broken down into parts using the sine and cosine functions for the specified direction angle \( \theta \). For example, using our given values of \( |\mathbf{u}| = 7 \) and \( \theta = 25^{\circ} \), we determine the component form:

• \( |\mathbf{u}| \cos(25^{\circ}) \approx 6.342 \)
• \( |\mathbf{u}| \sin(25^{\circ}) \approx 2.959 \)

So in component form, the vector is approximately \( \mathbf{u} = \langle 6.342, 2.959 \rangle \). This tells us exactly how the vector travels in the x and y directions.