Problem 56 - Trigonometry

Question

Assume vector \(\mathbf{V}\) is in standard position, has the given magnitude, and that \(\theta\) is the angle \(\mathbf{V}\) makes with the positive \(x\)-axis. Write \(\mathbf{V}\) in vector component form \(a \mathbf{i}+b \mathbf{j}\), and approximate your values to two significant digits. $$|\mathbf{V}|=8.5, \theta=97^{\circ}$$

Step-by-Step Solution

Answer

The vector in component form is \(-0.59 \mathbf{i} + 8.4 \mathbf{j}\).

Step 1: Understanding Vector Components

A vector in 2D space with a magnitude \(|\mathbf{V}|\) and angle \(\theta\) is expressed in component form as \(a \mathbf{i} + b \mathbf{j}\), where \(a\) and \(b\) are calculated as \(a = |\mathbf{V}| \cos(\theta)\) and \(b = |\mathbf{V}| \sin(\theta)\).

Step 2: Convert Angle to Radians

Convert \(\theta = 97^{\circ}\) into radians for use in trigonometric functions. The conversion is done using the formula \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \). Thus, \(97^{\circ} \approx 1.69 \text{ radians} \).

Step 3: Calculate the 'i' Component

Using the cosine function, calculate \(a\): \(a = 8.5 \cos(97^{\circ})\). Since \(97^{\circ} = 1.69 \text{ radians}\), \(a \approx 8.5 \cos(1.69) \approx -0.59\).

Step 4: Calculate the 'j' Component

Using the sine function, calculate \(b\): \(b = 8.5 \sin(97^{\circ})\). Since \(97^{\circ} = 1.69 \text{ radians}\), \(b \approx 8.5 \sin(1.69) \approx 8.4\).

Step 5: Write the Vector in Component Form

The vector \(\mathbf{V}\) in component form is \( -0.59 \mathbf{i} + 8.4 \mathbf{j} \) with values rounded to two significant digits.

Key Concepts

Vector Magnitude

When dealing with vectors, understanding the vector magnitude is important since it represents the length or size of the vector. In physics and mathematics, the magnitude is crucial for determining how large or intense a vector quantity is without considering its direction. This scalar quantity is often denoted by \(|\mathbf{V}|\), where \(\mathbf{V}\) is the vector itself.
For instance, a vector representing velocity will have a magnitude indicating how fast an object is moving. In our exercise, the vector magnitude given is 8.5.
To further conceptualize, envision drawing a vector on a graph: the magnitude would be equivalent to the length of the arrow irrespective of its direction.

Angle Conversion to Radians

Angles are crucial for expressing and calculating vector components, but they often need to be converted from degrees to radians to be used in trigonometric functions. The reason for this is that most mathematical functions, including trigonometric ones, require angles in radians rather than degrees.
The conversion formula is simple: multiply the angle in degrees by \(\frac{\pi}{180}\).
For example, an angle of 97° would translate to radians as follows:

• 97° \(\times\) \(\frac{\pi}{180}\) \(\approx 1.69\) radians.

This conversion is instrumental as it ensures trigonometric functions like sine and cosine work correctly in calculations.

Trigonometric Functions

Trigonometric functions such as sine and cosine are fundamental when breaking down vectors into components. These functions help relate the angle of the vector with its components along the x and y axes.
In the case of our vector problem, the sine function \(\sin(\theta)\) calculates the vertical component, while the cosine function \(\cos(\theta)\) finds the horizontal component.
Once the angle is converted to radians, these functions determine:

• \(a = |\mathbf{V}| \cos(\theta)\) for the 'i' component
• \(b = |\mathbf{V}| \sin(\theta)\) for the 'j' component

With \(\theta = 1.69\) radians in our exercise, you can see how each function extracts its respective vector component effectively, leading to our final values of \(-0.59\mathbf{i}\) and \(8.4\mathbf{j}\).

Vector Component Form

The final form of a vector in terms of its components gives a complete picture of both its magnitude and direction. It separates the vector into horizontal (x-axis) and vertical (y-axis) parts.
The vector component form is expressed as \(a \mathbf{i} + b \mathbf{j}\) in two dimensions, where \(a\) and \(b\) are the components calculated using the vector's magnitude and angle.
To summarize the process:

• Convert the angle to radians.
• Use trigonometric functions to find \(a\) and \(b\).
• Combine these components in the form \(-0.59 \mathbf{i} + 8.4 \mathbf{j}\).

This form is highly practical as it allows anyone to visualize and work with the vector in a two-dimensional plane clearly. It ensures clarity in how a vector behaves spatially, especially in physics and engineering contexts.