Problem 5 - Technical Mathematics with Calculus

Question

Given the magnitude of each vector and the angle \(\theta\) that it makes with the \(x\) axis, find the \(x\) and \(y\) components. $$\text { Magnitude }=836 \quad \theta=45.2^{\circ}$$

Step-by-Step Solution

Answer

The x-component is approximately 836 * cos(45.2^\circ) and the y-component is approximately 836 * sin(45.2^\circ).

Step 1: Understanding the Vector Components

To find the x and y components of a vector, one can use the trigonometric relationships involving the magnitude of the vector and the angle it makes with the x-axis. The x-component is found by multiplying the magnitude by the cosine of the angle, and the y-component is found by multiplying the magnitude by the sine of the angle.

Step 2: Calculating the X-Component

Calculate the x-component (horizontal component) of the vector by using the formula: x-component = magnitude * cos(\(\theta\)). Since the magnitude of the vector is 836 and \(\theta = 45.2^\circ\), the calculation will be x-component = 836 * cos(45.2^\circ). Use a calculator to find the cosine of 45.2 degrees and then multiply by the magnitude.

Step 3: Calculating the Y-Component

Calculate the y-component (vertical component) of the vector by using the formula: y-component = magnitude * sin(\(\theta\)). Again, using the magnitude of 836 and \(\theta = 45.2^\circ\), the calculation will be y-component = 836 * sin(45.2^\circ). Find the sine of 45.2 degrees using a calculator and multiply by the magnitude.

Key Concepts

Trigonometric Relationships for Vectors

Understanding the trigonometric relationships for vectors is crucial when trying to break down a vector into its components. This process is a bit like dismantling a toy to see what parts it is made of. Imagine a vector as an arrow that has both direction and magnitude. The angle this arrow makes with the horizontal line (x-axis) helps determine its direction.

To find the components along the x and y axes, we use two fundamental trigonometric functions: sine (sin) and cosine (cos). These functions relate the angle of the vector to its x and y components. For a vector at an angle \(\theta\) from the x-axis, the cosine of that angle, \(\cos(\theta)\), gives the ratio of the x-component (horizontal) to the vector's magnitude. Similarly, the sine of the angle, \(\sin(\theta)\), corresponds to the ratio of the y-component (vertical) to the magnitude.

Think of it as asking two questions: 'How much does the vector go right?' which is answered by the cosine, and 'How much does it go up?' answered by the sine. This relationship is straightforward yet powerful, allowing us to resolve a slanted movement into horizontal and vertical parts that are easier to work with.

Calculating Vector Magnitude

When it comes to vectors, magnitude represents how strong or how far the vector goes, without caring for its direction. It's like knowing how fast a car is going without knowing where it is headed. Calculating the magnitude is essential when we work backward from components to find out the overall strength of the vector.

If you already have the x and y components, you can imagine forming a right triangle where these components are the legs, and the vector itself is the hypotenuse. To find the magnitude, which is the length of the hypotenuse, we employ the Pythagorean theorem: \( magnitude = \sqrt{x-component^2 + y-component^2} \).

This formula is quite helpful because, even if you only have the components, you can still determine how strong the vector is. It's a bit like knowing the size of a TV screen by measuring its width and height.

X and Y Components of a Vector

Splitting a vector into its x (horizontal) and y (vertical) components is like giving directions using a grid system instead of a compass. It's a way to describe movement in a two-dimensional space using two perpendicular lines, the x-axis and y-axis.

To extract these components from a vector, we apply the trigonometric relationships we've discussed. The x-component, often called the horizontal component, is found by multiplying the vector's magnitude by the cosine of the angle it forms with the x-axis: \(\text{x-component} = \text{magnitude} \cdot \cos(\theta)\).

Conversely, the y-component, or the vertical one, is derived by multiplying the magnitude by the sine of the angle: \(\text{y-component} = \text{magnitude} \cdot \sin(\theta)\).

Finding Components in Practice

Let's consider an example: if a vector has a magnitude of 836 and makes a 45.2° angle with the x-axis, the x-component will be found by multiplying 836 by the cosine of 45.2°. Similarly, the y-component is calculated by multiplying 836 by the sine of 45.2°. These components are crucial when we need to work with vectors in different directions, such as calculating the result when two vectors are added together or figuring out the impact force in a physical simulation.