Question

Find the resultant of each pair of vectors. $$83.2 \underline{/ 49.7^{\circ}} \text { and } 52.5 \underline{/ 66.3^{\circ}}$$

Short answer

The resultant vector can be found by first converting the given vectors to rectangular components, summing these components, and then converting the sum back to polar form.

Step-by-step solution

- Convert Polar Vectors to Rectangular Form

Convert both vectors from polar form (magnitude and angle) to their rectangular form (x and y components) using the following formula: For a vector with magnitude 'r' and angle '\(\theta\)', the x-component is '\(r \cos(\theta)\)' and the y-component is '\(r \sin(\theta)\)'. The first vector: \(x_1 = 83.2 \cos(49.7^\circ)\), \(y_1 = 83.2 \sin(49.7^\circ)\). The second vector: \(x_2 = 52.5 \cos(66.3^\circ)\), \(y_2 = 52.5 \sin(66.3^\circ)\).

- Compute the Rectangular Components

Calculate the x and y components for both vectors using the cosine and sine of their respective angles: First vector components: \(x_1 = 83.2 \cos(49.7^\circ)\) and \(y_1 = 83.2 \sin(49.7^\circ)\). Second vector components: \(x_2 = 52.5 \cos(66.3^\circ)\) and \(y_2 = 52.5 \sin(66.3^\circ)\).

- Sum the Rectangular Components of Both Vectors

To find the resultant vector, sum the x and y components of both vectors. Resultant x-component \(R_x = x_1 + x_2\). Resultant y-component \(R_y = y_1 + y_2\).

- Convert the Resultant Vector Back to Polar Form

After obtaining the resultant rectangular components, convert them back to polar form. Use the formula \(R = \sqrt{R_x^2 + R_y^2}\) for the magnitude and \(\theta = \text{atan2}(R_y, R_x)\) for the angle, which accounts for the correct quadrant of the resultant vector.

Key concepts

Polar to Rectangular Conversion

Understanding the conversion from polar to rectangular (or Cartesian) coordinates is key when working with vectors, especially in physics and engineering fields. In polar form, a vector is described by its magnitude, which tells us how long the vector is, and its angle, which tells us the direction it's pointing. To convert these to rectangular form, we break the vector down into horizontal (x) and vertical (y) components. This is done using trigonometry:

To find the x-component of a vector, you multiply the magnitude by the cosine of the angle: \( x = r \cos(\theta) \). Similarly, the y-component is found by multiplying the magnitude by the sine of the angle: \( y = r \sin(\theta) \). These components correspond to the distances along the horizontal and vertical axes in a two-dimensional coordinate system, creating a right triangle with the vector itself as the hypotenuse.

• x-component (horizontal): \( x = r \cos(\theta) \)
• y-component (vertical): \( y = r \sin(\theta) \)

This conversion is crucial because it allows us to easily perform vector addition, which is the base for our problem's solution.

Rectangular Components Calculation

Once we have our vector components in rectangular form, the next step is to calculate these components for vector addition. This process involves determining the individual horizontal and vertical distances each vector contributes to the overall direction. For each vector, the x and y components are computed separately.

For instance, as seen in the solution for the provided vectors, one would calculate the first vector's components as such:

First Vector Components:

• x-component: \( x_1 = 83.2 \cos(49.7^\circ) \)
• y-component: \( y_1 = 83.2 \sin(49.7^\circ) \)

And for the second vector:

Second Vector Components:

• x-component: \( x_2 = 52.5 \cos(66.3^\circ) \)
• y-component: \( y_2 = 52.5 \sin(66.3^\circ) \)

With these calculations, each vector is broken down into a pair of values that represent the steps you would take along the x (horizontal) and y (vertical) axes to follow the direction of the original vector. These steps make adding vectors together a matter of simple arithmetic.

Resultant Vector in Polar Form

After summing the x and y components separately, we have the rectangular form of the resultant vector. But often, we want to express this as a single magnitude and direction or in other words, in polar form.

To get back to polar coordinates, we need to calculate the magnitude and angle of the resultant vector. The magnitude of the resultant vector \( R \) is given by the square root of the sum of the squares of its components: \( R = \sqrt{R_x^2 + R_y^2} \). This is essentially the Pythagorean theorem, as our x and y components form a right triangle with the resultant vector.

The angle \( \theta \) of the resultant vector is a bit trickier, as we need to take into account which quadrant the resultant vector lies in. This is done using the 'atan2' function: \( \theta = \text{atan2}(R_y, R_x) \). This function computes the arc tangent of the quotient \( \frac{R_y}{R_x} \) while maintaining the correct sign for the numbers, allowing it to cover more than just the standard range of the regular arc tangent function (-90° to +90°), thereby providing the angle in the full 360° range.

By converting to polar form, we can easily visualize and communicate the combined effect of the original vectors, making it a very practical and beneficial process in fields such as physics, navigation, and robotics.