How To Find The Resultant of Two Vectors

Summary

This video explains how to calculate the magnitude and direction of the resultant vector formed by the sum of two force vectors. It demonstrates how to break down each force vector into its x and y components using trigonometric functions. After finding the components of both vectors, they are added together to form the resultant vector. The video walks through the process of calculating the magnitude using the Pythagorean theorem and finding the reference angle through arc tangent. The final result provides both the magnitude and the direction of the resultant vector relative to the positive x-axis.

Takeaways

• 😀 To calculate the resultant vector, break down each force vector into its x and y components.
• 😀 The x-component of a force vector is calculated by multiplying the force by the cosine of the angle (f1x = f1 * cos(θ)).
• 😀 The y-component of a force vector is calculated by multiplying the force by the sine of the angle (f1y = f1 * sin(θ)).
• 😀 A unit vector is a vector with a magnitude of 1, and is used to represent the x and y components (i for x-axis, j for y-axis).
• 😀 The x and y components of force vectors are added to find the resultant vector in component form.
• 😀 If the x-component of a vector is negative, it indicates the vector is in the second or third quadrant (i.e., opposite direction).
• 😀 The resultant vector is the sum of the individual x and y components of all vectors.
• 😀 The magnitude of the resultant vector is calculated using the Pythagorean theorem: sqrt(a² + b²).
• 😀 To find the direction of the resultant vector, the reference angle can be determined using the arc tangent function (tan⁻¹(y/x)).
• 😀 Once the reference angle is found, the direction relative to the positive x-axis can be determined by subtracting the reference angle from 180 degrees if needed.
• 😀 In this example, the resultant vector has a magnitude of 314.5 newtons and a direction of 97.1 degrees counter-clockwise from the positive x-axis.

Q & A

What is the first step in finding the resultant vector of two force vectors?

The first step is to break down each force vector into its component form, which involves determining the x and y components for each vector.

How is the x-component of a force vector calculated?

The x-component of a force vector is calculated by multiplying the magnitude of the force by the cosine of the angle, i.e., f1x = f1 * cos(theta).

What role do the unit vectors i and j play in the component form of a vector?

The unit vectors i and j represent the direction along the x and y axes respectively. A force vector in component form is expressed as a sum of these components, e.g., f1 = 173.2i + 100j.

Why is the x-component of the second vector negative?

The x-component of the second vector is negative because the vector lies in the second quadrant, where the x-values are negative, resulting in a negative value for the x-component.

How do you calculate the y-component of the second force vector?

The y-component of the second force vector is calculated by multiplying the magnitude of the force by the sine of the angle, i.e., f2y = f2 * sin(theta), which in this case gives a positive value.

What is the process for finding the resultant vector in component form?

To find the resultant vector in component form, add the x-components and the y-components of the individual vectors together. For example, the resultant x-component is f1x + f2x, and the resultant y-component is f1y + f2y.

How do you calculate the magnitude of the resultant vector?

The magnitude of the resultant vector is calculated using the Pythagorean theorem: the square root of the sum of the squares of the x and y components, i.e., magnitude = sqrt(fx^2 + fy^2).

What does the negative sign in the x-component of the resultant vector indicate?

The negative sign in the x-component of the resultant vector indicates that the vector is pointing to the left of the positive x-axis, in the negative x-direction.

How is the reference angle of the resultant vector calculated?

The reference angle of the resultant vector is calculated by taking the arctangent of the ratio of the y-component to the x-component, i.e., angle = arctan(fy / fx), and then adjusting the angle based on the quadrant.

How do you find the direction of the resultant vector relative to the positive x-axis?

The direction of the resultant vector relative to the positive x-axis is found by subtracting the reference angle from 180 degrees if the vector is in the second quadrant, as in this case.