Question

Find the resultant of each set of vectors. $$284 \underline{/ 33.6^{\circ}} \text { and } 184 \underline{/ 11.5^{\circ}}$$

Short answer

\( R_x \approx 284 \times \text{cos}(33.6^\circ) + 184 \times \text{cos}(11.5^\circ) \), \( R_y \approx 284 \times \text{sin}(33.6^\circ) + 184 \times \text{sin}(11.5^\circ) \), then \( R \approx \sqrt{R_x^2 + R_y^2} \) and \( \theta_R \approx \text{atan2}(R_y, R_x) \).

Step-by-step solution

Break Down Vectors into Components

Express each vector in terms of its horizontal (x) and vertical (y) components using trigonometric functions. For a vector with magnitude R and angle θ, you can calculate the components using: \( R_x = R \times \text{cos}(θ) \) and \( R_y = R \times \text{sin}(θ) \).

Calculate Components of the First Vector

Calculate the x and y components of the first vector with magnitude 284 units and angle 33.6 degrees: \( x_1 = 284 \times \text{cos}(33.6^\circ) \) and \( y_1 = 284 \times \text{sin}(33.6^\circ) \).

Calculate Components of the Second Vector

Calculate the x and y components of the second vector with magnitude 184 units and angle 11.5 degrees: \( x_2 = 184 \times \text{cos}(11.5^\circ) \) and \( y_2 = 184 \times \text{sin}(11.5^\circ) \).

Sum the Components

Add the corresponding components of both vectors to get the resultant vector's components: \( R_x = x_1 + x_2 \) and \( R_y = y_1 + y_2 \).

Calculate the Magnitude of the Resultant Vector

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

Calculate the Direction of the Resultant Vector

Use the arctangent function to calculate the angle (direction) of the resultant vector: \( \theta_R = \text{atan2}(R_y, R_x) \), which considers the signs of both components to determine the correct quadrant.

Key concepts

Vector Components

Understanding vector components is crucial when dealing with physics problems involving vectors. A vector is a quantity that has both magnitude and direction, and it can be broken down into two parts: the horizontal (x-axis) and vertical (y-axis) components. This breakdown is especially useful when adding vectors that are not aligned along the same direction.

To find these components, we use basic trigonometry: the cosine function gives us the x-component, and the sine function gives us the y-component. So, for a vector with magnitude R and angle θ from the positive x-axis, its components can be calculated as R_x = R × cos(θ) and R_y = R × sin(θ).

By breaking down vectors into components, we simplify the problem into manageable parts that follow the rules of scalar addition, which is much easier than trying to directly add vectors at various angles.

Trigonometric Functions in Vectors

The use of trigonometric functions in vector calculations allows us to transition from polar coordinates (using magnitude and angle) to rectangular coordinates (using x and y components). Specifically, in the context of vectors, the cosine and sine functions help us calculate the horizontal and vertical components, respectively.

Cosine and Vectors

The cosine of an angle in a right-angled triangle describes the ratio of the adjacent side to the hypotenuse. In vector calculations, we use the cosine to determine the length of the vector's projection along the x-axis.

Sine and Vectors

Similarly, the sine function provides the ratio of the opposite side to the hypotenuse. This is used to find the length of the vector's projection along the y-axis. It's the combination of these two trigonometric functions that allows us to fully express the vector in terms of its components.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics that states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is widely used in vector analysis to determine the magnitude of a resultant vector.

Once we have calculated the components of the resultant vector using the x and y axes as references, we can treat these as the two smaller sides of a right-angled triangle, with the resultant vector being the hypotenuse. Mathematically, if R_x and R_y are the x and y components of the resultant vector, the magnitude R is given by R = √(R_x² + R_y²).

This formula is essential for converting the rectangular form of the vector back into polar form, giving us the overall size (magnitude) of the resultant vector, which can then be used along with trigonometric functions to find its direction.