Question

Find the resultant of each pair of vectors. $$4.83 \underline{/ 18.3^{\circ}} \text { and } 5.99 \underline{/ 83.5^{\circ}}$$

Short answer

The magnitude of the resultant vector is \(R = \sqrt{(4.83 \cdot \cos(18.3^\circ) + 5.99 \cdot \cos(83.5^\circ))^2 + (4.83 \cdot \sin(18.3^\circ) + 5.99 \cdot \sin(83.5^\circ))^2}\), and the direction is \(\theta = \text{atan2}(4.83 \cdot \sin(18.3^\circ) + 5.99 \cdot \sin(83.5^\circ), 4.83 \cdot \cos(18.3^\circ) + 5.99 \cdot \cos(83.5^\circ))\).

Step-by-step solution

Draw the Vectors

Start by drawing both vectors on a coordinate system. The first vector 4.83 has a magnitude of 4.83 units and makes an angle of 18.3 degrees with the positive x-axis. The second vector 5.99 has a magnitude of 5.99 units and makes an angle of 83.5 degrees with the positive x-axis.

Resolve the Vectors into Components

Resolve each vector into its horizontal (x) and vertical (y) components. For vector 1 with magnitude 4.83 and angle 18.3 degrees, the components are: \(x_1 = 4.83 \cdot \cos(18.3^\circ)\) and \(y_1 = 4.83 \cdot \sin(18.3^\circ)\). For vector 2 with magnitude 5.99 and angle 83.5 degrees, the components are: \(x_2 = 5.99 \cdot \cos(83.5^\circ)\) and \(y_2 = 5.99 \cdot \sin(83.5^\circ)\).

Calculate the Component Values

Compute the values of the x and y components for both vectors using a calculator set to degrees: \(x_1 = 4.83 \times \cos(18.3^\circ)\), \(y_1 = 4.83 \times \sin(18.3^\circ)\), \(x_2 = 5.99 \times \cos(83.5^\circ)\), and \(y_2 = 5.99 \times \sin(83.5^\circ)\).

Add the Components Together

Find the resultant vector's components by adding the corresponding components of the two vectors: \(x_{\text{resultant}} = x_1 + x_2\) and \(y_{\text{resultant}} = y_1 + y_2\).

Calculate the Resultant Vector

Compute the magnitude of the resultant vector using the Pythagorean theorem: \(R = \sqrt{x_{\text{resultant}}^2 + y_{\text{resultant}}^2}\), and find the direction by calculating the angle with the positive x-axis: \(\theta = \text{atan2}(y_{\text{resultant}}, x_{\text{resultant}})\) using the arctangent function which takes into account the signs of the components to determine the correct quadrant.

Key concepts

Resultant Vector

Imagine you're on a treasure hunt, you go 4.83 units northeast, then head 5.99 units more in a direction closer to north. Your actual position compared to where you started is like finding the resultant vector of your journey. It combines both moves into a single, direct path to the treasure.

In physics and mathematics, this concept helps us to understand movement, forces, or any quantities that have both magnitude and direction. Think of it like adding arrows end-to-end; the resultant vector is the arrow that links the start to the end point directly. It simplifies the situation and shows the overall effect of combined vectors.

Vector Components

Vector components are essentially pieces of the original vector, split along perpendicular axes, usually the x (horizontal) and y (vertical) axes. If we metaphorically slice a vector diagonally, these components are the shadows it would cast on the two axes. Each component reflects the influence of the vector in that particular direction.

To find a vector's components, you use trigonometry: for a vector with magnitude M and angle θ, the horizontal component is M × cos(θ) and the vertical component is M × sin(θ). This process is like unwinding a twisted rope into straight lines along the x and y axes, making it easier to work with.

Magnitude and Direction of Vectors

Just as you describe a flight's journey by its distance and where it's headed, vectors are characterized by their magnitude (how long they are) and direction (the way they're pointing). You wouldn't find your flight on length alone, and the same goes with vectors: both properties are crucial.

The magnitude tells us 'how much' or 'how strong' and is always a non-negative number, while the direction gives us the 'which way'. When combining vectors, the magnitude and direction of the resultant vector give us a complete picture of the combined effect.

Trigonometry in Vector Addition

We often use trigonometry, the study of triangles, like a Swiss Army knife for vector addition. Sin, cos, and tan are like different blades we use to dissect vectors into manageable parts.

• If a vector points in a direction not aligned with our axes, trig functions help break it down.
• Cosine gets the length along the x-axis while sine fetches how far it goes up or down along the y-axis.
• Once we have components, we can add vectors by simply combining these values along the corresponding axes.
• To go full circle, we use inverse trigonometry (like arctan) to find the angle of the resultant vector, retracing our steps back into the real world direction.

Tools like Pythagorean theorem also join the party for finding the magnitude of the resultant vector, cementing trigonometry as our go-to helper in vector addition.