Problem 42 - Precalculus : Building Concepts and Connections

Question

Find the components of the vector in standard position that satisfy the given conditions. Length \(7 ;\) direction \(276^{\circ}\)

Step-by-Step Solution

Answer

To find the vector's components in standard position, apply the formulas for vector to Cartesian conversion. This means x-component equals \(7 \cos (276^{\circ})\) and y-component equals to \(7 \sin (276^{\circ})\). Convert these to decimal values (if necessary) and present them as an ordered pair (i, j).

Step 1: Compute x-component (i) of the vector

We'll start by determining the x component (i) of the vector. The formula for this is \(R \cos\theta\), where R represents the vector's length and \(\theta\) is the direction angle. Given that R = 7 and \(\theta = 276^{\circ}\), substituting these values gives us: \(i = 7 \cos (276^{\circ})\).

Step 2: Compute y-component (j) of the vector

Next, we'll determine the y component (j) of the vector. The formula for this is \(R \sin\theta\) where, as before, R represents the vector's length and \(\theta\) is the direction angle. Given that R = 7 and \(\theta = 276^{\circ}\), substituting these values gives us: \(j = 7 \sin (276^{\circ})\).

Step 3: Write the Co-ordinates in 2D Cartesian Form

The vector, in standard position and 2D Cartesian form, follows this notation format: \((i, j)\). Substituting the calculated values of i (from Step 1) and j (from Step 2) provides the vector components in standard position.

Key Concepts

Vector Components

Vectors are fundamental in physics and engineering, representing quantities with both magnitude and direction. When dealing with vectors, understanding their components is key. A vector in a plane can be split into two parts: the x-component and the y-component. These components identify how far the vector stretches along the x-axis and the y-axis, respectively.
To find the components, use trigonometric functions:

• The x-component (i) is given by: \( R \cos \theta \)
• The y-component (j) is given by: \( R \sin \theta \)

Here, \( R \) represents the length of the vector, and \( \theta \) is the angle of direction. Together, these components form a complete picture of the vector's influence in a 2D space.

2D Cartesian Form

The 2D Cartesian form is a simple way to express vectors using coordinates in a plane. It's like giving an address to the vector in the 2D world. Imagine a graph with x and y axes; the vector resides at point \((i, j)\) on this graph.
This form makes it easier to visualize and calculate vectors:

• The x-coordinate corresponds to the horizontal reach of the vector.
• The y-coordinate corresponds to the vertical reach of the vector.

In a real-world scenario, this helps understand where the vector points and how it interacts with other vectors. It becomes immensely useful in areas like navigation and physics.

Trigonometric Functions

Trigonometric functions like cosine and sine are essential in breaking down vectors into components. These functions relate the angles and sides of right triangles, which is why they perfectly suit vector calculations.
Consider a vector with angle \( \theta \) and length \( R \):

• Cosine \( (\cos) \) helps find the x-component, showing how much the vector extends horizontally.
• Sine \( (\sin) \) helps find the y-component, showing how much the vector extends vertically.

By using these functions, you can easily analyze and manipulate vectors even without directly knowing their graphical appearance.

Vector Length

The vector length, also known as magnitude, tells us how strong or large the vector is. Think of it like the length of an arrow in a diagram, indicating the vector's impact.
Represented by \( R \), the vector length can be calculated if components are known, using the formula:\[ R = \sqrt{i^2 + j^2} \]Here, \( i \) and \( j \) are the x and y components. In scenarios where the vector is represented by its angle and length, understanding this helps derive components, paving the way for further analysis.

Direction Angles

Direction angles describe the orientation of vectors in a plane, providing insight into where the vector points. They are measured counterclockwise from the positive x-axis.
In our example, a direction angle of \( 276^{\circ} \) shows that the vector points predominantly downward and slightly to the left. This angle helps determine the sign and value of the components using trigonometric functions.

• Angles between \( 0^{\circ} \) and \( 360^{\circ} \) map the entire plane.
• Commonly, angles exceeding \( 90^{\circ} \) often involve negative x or y components, fostering understanding of the vector's path.

Understanding direction angles is crucial for accurately placing vectors within a given coordinate system.