Problem 30 - College Trigonometry

Question

Approximate the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Round your approximations to two decimal places. \(\|\vec{v}\|=168.7 ;\) when drawn in standard position \(\vec{v}\) makes a \(252^{\circ}\) angle with the positive \(x\) -axis

Step-by-Step Solution

Answer

\(\vec{v} \approx \langle -52.11, -160.51 \rangle\)

Step 1: Understand the Problem

We need to find the component form of the vector \(\vec{v}\), which has a given magnitude \(\|\vec{v}\|\) and direction (angle with the positive x-axis). The component form is expressed as \(\vec{v} = \langle v_x, v_y \rangle\), where \(v_x\) and \(v_y\) are the components along the x-axis and y-axis, respectively.

Step 2: Use Magnitude and Direction to Find Components

The component formulas for a vector given its magnitude and direction are: \[ v_x = \|\vec{v}\| \cdot \cos(\theta) \]\[ v_y = \|\vec{v}\| \cdot \sin(\theta) \]In this case, \(\theta = 252^{\circ}\).

Step 3: Calculate the x-component

Substitute the given values into the formula for the x-component:\[ v_x = 168.7 \cdot \cos(252^{\circ}) \]Calculate using a calculator (ensure it is in degree mode):\[ v_x \approx 168.7 \times (-0.309) \approx -52.11 \]

Step 4: Calculate the y-component

Substitute the given values into the formula for the y-component:\[ v_y = 168.7 \cdot \sin(252^{\circ}) \]Calculate using a calculator:\[ v_y \approx 168.7 \times (-0.951) \approx -160.51 \]

Step 5: Write the Component Form

Combine the x and y components to write the vector in component form:\[ \vec{v} = \langle -52.11, -160.51 \rangle \]

Key Concepts

Magnitude and Direction of a Vector

When working with vectors, understanding their magnitude and direction is key. The magnitude refers to the size or length of the vector. It's a scalar value that tells us how long the vector is. You can think of it as the vector's equivalent of the length of a line segment.
Direction, on the other hand, describes where the vector points. This is typically measured as an angle from a reference direction, which is often the positive x-axis. The given angle, such as the 252° in our exercise, shows the amount of rotation from the positive x-axis to our vector.
In summary, the magnitude tells us "how much", while the direction tells us "where". Our goal in problems like these is to use both pieces of information to find the exact position of the vector in a coordinate system.

Using Trigonometric Functions to Find Vector Components

Trigonometric functions, like sine and cosine, are essential in determining the components of a vector given its magnitude and direction. By decomposing a vector into horizontal and vertical components, you align it with a coordinate grid.
For a vector \( \vec{v} \), the formulas for its horizontal component, \( v_x \), and vertical component, \( v_y \), use cosine and sine, respectively:

• \( v_x = \|\vec{v}\| \cdot \cos(\theta) \)
• \( v_y = \|\vec{v}\| \cdot \sin(\theta) \)

The angle \( \theta \) used in these formulas should be in relation to the positive x-axis. For example, if \( \theta = 252^{\circ} \), it means the vector is pointing in a specific direction within the coordinate plane.
By substituting the angle into these functions, you can find \( v_x \) and \( v_y \), which tell you how far and in which direction the vector moves along the x and y axes.

Vector Notation for Representing Vectors

Vector notation is a systematic way to write vectors clearly and concisely. In two-dimensional space, we typically use angle brackets \( \langle v_x, v_y \rangle \) to denote vectors, where \( v_x \) and \( v_y \) are the vector's components.
This notation highlights the fact that vectors have both direction and magnitude. The components \( v_x \) and \( v_y \) indicate how much the vector moves along the x-axis and y-axis, respectively.
For instance, in our exercise, the vector is represented as \( \vec{v} = \langle -52.11, -160.51 \rangle \). This tells us:

• The vector moves -52.11 units along the x-axis.
• It moves -160.51 units along the y-axis.

By using this simple notation, vectors can be accurately described, calculated, and communicated, making it an essential tool in mathematics and physics.