Problem 57 - 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. $$|\mathbf{V}|=0.55, \theta=195^{\circ}$$

Step-by-Step Solution

Answer

\( \mathbf{V} \approx -0.53 \mathbf{i} - 0.14 \mathbf{j} \)

Step 1: Understand the Vector Components

To write a vector \( \mathbf{V} \) in component form \( a \mathbf{i} + b \mathbf{j} \), we need to determine its horizontal component \( a \) and vertical component \( b \). These components are related to the vector's magnitude and its angle from the positive \( x \)-axis.

Step 2: Calculate the Horizontal Component

The horizontal component \( a \) of vector \( \mathbf{V} \) is found using the formula: \[ a = |\mathbf{V}| \cdot \cos(\theta) \]Substitute the values given: \[ a = 0.55 \cdot \cos(195^{\circ}) \].Using a calculator, evaluate \( \cos(195^{\circ}) \).

Step 3: Calculate \(\cos(195^{\circ})\)

The cosine of 195 degrees can be calculated or looked up from a trigonometric table: \[ \cos(195^{\circ}) = -0.9659 \].

Step 4: Determine \(a\) Using Calculated Cosine

Now calculate the horizontal component using the cosine value: \[ a = 0.55 \cdot (-0.9659) = -0.531245 \].Round it to two significant digits: \[ a \approx -0.53 \].

Step 5: Calculate the Vertical Component

The vertical component \( b \) of vector \( \mathbf{V} \) is found using the formula: \[ b = |\mathbf{V}| \cdot \sin(\theta) \]Substitute the given values:\[ b = 0.55 \cdot \sin(195^{\circ}) \].

Step 6: Calculate \(\sin(195^{\circ})\)

The sine of 195 degrees can be calculated or looked up from a trigonometric table: \[ \sin(195^{\circ}) = -0.2588 \].

Step 7: Determine \(b\) Using Calculated Sine

Calculate the vertical component using the sine value: \[ b = 0.55 \cdot (-0.2588) = -0.14234 \].Round it to two significant digits: \[ b \approx -0.14 \].

Step 8: Write the Vector in Component Form

Combine the calculated components to express the vector in component form: \[ \mathbf{V} \approx -0.53 \mathbf{i} - 0.14 \mathbf{j} \].

Key Concepts

Magnitude of a Vector

The magnitude of a vector, often denoted as \(|\mathbf{V}|\), indicates the length or size of the vector. It is a scalar quantity, which means it has only magnitude and no direction. This is similar to measuring the distance a point is from the origin in the vector plane. In the given exercise, the magnitude of the vector \(\mathbf{V}\) is provided as 0.55. This means that if you were to think of the vector as a directed arrow starting at the origin, the arrow would stretch 0.55 units in the plane.
The magnitude plays a crucial role in determining the components of a vector. Each component is scaled by the magnitude depending on the vector's direction. Understanding the magnitude helps us ascertain how far in the space the vector extends and is necessary for calculating directional components using trigonometric functions.

Angle with Positive x-axis

The angle \(\theta\) that a vector makes with the positive \(x\)-axis is a measure of direction from the horizontal. It is measured in degrees or radians from the positive direction of the \(x\)-axis towards the vector. In our context, the vector \(\mathbf{V}\) makes an angle of 195 degrees with the positive \(x\)-axis. This implies the vector is directed more towards the left in a coordinate plane, as 195 degrees lies in the third quadrant.
Calculating components from this angle requires using the angle in trigonometric functions (cosine for the horizontal component and sine for the vertical component). Remembering that standard trigonometric functions consider their inputs as an angle from the positive \(x\)-axis helps you correctly find the direction of the vector’s components.

Trigonometric Functions

Trigonometric functions are essential in breaking down vectors into their components. Specifically, cosine (\(\cos\)) and sine (\(\sin\)) functions aid in translating an angle into horizontal and vertical components. Let's go through how each function is used:

• Cosine Function: Used to determine the horizontal component of the vector. For vector \(\mathbf{V}\), this involves the calculation \( a = |\mathbf{V}| \cdot \cos(\theta) \), which gives the dimension in the \(x\)-direction.
• Sine Function: Utilized for the vertical component. The calculation \( b = |\mathbf{V}| \cdot \sin(\theta) \) provides the dimension in the \(y\)-direction.

In the exercise, the use of a calculator is encouraged to find \(\cos(195^{\circ})\) and \(\sin(195^{\circ})\), resulting in values \(-0.9659\) and \(-0.2588\) respectively. These values indicate that both components are negative, confirming the vector lies in the third quadrant of the Cartesian plane, where both \(x\) and \(y\) values are negative.