Question

Find the resultant of each set of vectors. $$735 \underline{/ 28.4^{\circ}}, 68.4 \underline{/ 22.5^{\circ}}, 33.7 \underline{/ 52.4^{\circ}}, \text { and } 483 \underline{/ 44.8^{\circ}}$$

Short answer

To find the resultant vector, compute the rectangular components for each given vector, sum the components, and then calculate the magnitude and direction of the resultant vector.

Step-by-step solution

Represent Each Vector in Component Form

Convert each polar vector to its rectangular components using the formulas: \( x = r \times \text{cos}(\theta) \) and \( y = r \times \text{sin}(\theta) \), where \( r \) is the magnitude and \( \theta \) is the direction of the vector.

Calculate the Rectangular Components for Each Vector

For each vector, calculate the x (horizontal) component and the y (vertical) component using the given magnitude and angle. For example, for the first vector \( 735 \text{ at } 28.4^{\text{o}} \), the components are: \( x_1 = 735 \times \text{cos}(28.4^{\text{o}}) \) and \( y_1 = 735 \times \text{sin}(28.4^{\text{o}}) \). Perform the same calculations for the other three vectors using their respective magnitudes and angles.

Sum up the Components

Add up all the x-components to find \( R_x \), the resultant vector's x-component. Similarly, sum all the y-components to find \( R_y \), the resultant vector's y-component.

Determine the Magnitude of the Resultant Vector

Calculate the magnitude of the resultant vector using the Pythagorean theorem: \( R = \sqrt{R_x^2 + R_y^2} \).

Determine the Angle of the Resultant Vector

Compute the angle of the resultant vector with respect to the positive x-axis using the formula: \( \theta_R = \text{atan2}(R_y, R_x) \), where \( \text{atan2} \) is the two-argument arctangent function which takes into account the signs of both vector components to return the correct quadrant.

Key concepts

Component Form of Vectors

The component form of vectors is a method to express a vector by separating it into its horizontal (x-component) and vertical (y-component) parts. This is particularly useful when dealing with vectors in a two-dimensional plane.

For a vector given in polar coordinates, where the magnitude (r) and angle (θ) are known, the x and y components can be found using trigonometric functions. The x-component is given by the product of the magnitude and the cosine of the angle \( x = r \times \cos(\theta) \), while the y-component is given by the product of the magnitude and the sine of the angle \( y = r \times \sin(\theta) \).

Why the Component Form Matters

Working in the component form simplifies many vector operations, such as addition, subtraction, and even finding the resultant. By breaking down vectors into components, calculation becomes a matter of simple arithmetic rather than complex geometry.

Polar to Rectangular Conversion

Polar to rectangular conversion is a method used to switch between two ways of describing a point or a vector in a plane. In polar form, a point is described by its distance from the origin (magnitude) and the angle it forms with the positive x-axis (direction).

In contrast, the rectangular form describes a point based on its horizontal and vertical displacement from the origin, represented as (x, y) coordinates. To convert a vector from polar to rectangular form, you use the relationships \( x = r \times \cos(\theta) \) and \( y = r \times \sin(\theta) \).

Visualizing the Conversion

Imagine drawing a right triangle with the vector forming the hypotenuse. The adjacent side of the triangle is the x-component, while the opposite side is the y-component. Trigonometry leads us from the polar coordinates to the precise (x, y) coordinates of the rectangular form.

Vector Addition

Vector addition involves combining two or more vectors to find a single vector called the resultant. This operation is based on the head-to-tail method or, more easily, by adding corresponding components.

To add vectors in component form, simply add all the x-components together to produce the x-component of the resultant, and add all the y-components together for the resultant's y-component. So, if you have vectors with components \( (x_1, y_1) \) and \( (x_2, y_2) \) the resultant vector \( R \) will have the components \( R_x = x_1 + x_2 \) and \( R_y = y_1 + y_2 \).

The Essence of Vector Addition

It's like adding distances in the east-west direction (x-components) and the north-south direction (y-components) separately. The idea is to find the total displacement, represented by the resultant vector, which shows the combined effect of both vectors.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Expressed as an equation, it is \( c^2 = a^2 + b^2 \), where c represents the length of the hypotenuse, and a and b represent the lengths of the triangle's other two sides.

In the context of vectors, this theorem helps to calculate the magnitude of the resultant vector in component form. After adding the x and y components of the vectors using vector addition, the magnitude of the resultant \( R \) is \( R = \sqrt{R_x^2 + R_y^2} \). Here, \( R_x \) and \( R_y \) are analogous to a and b in the Pythagorean theorem, and the magnitude R is like c.

Applying the Pythagorean Theorem

If you visualize the components as sides of a right-angled triangle, the resultant vector's magnitude corresponds to the length of the hypotenuse. So, this theorem provides the final step to move from components back to the polar form, fully describing the vector's effect.