# Resultant of 3 vectors along the sides of an equilateral triangle

• cr7einstein
In summary, the problem involves finding the resultant of three parallel forces with magnitudes of 10N, 20N, and 30N acting on a point in the directions of the sides of an equilateral triangle. The answer cannot be zero as the triangle only represents the direction of the forces, not their magnitudes. The correct method involves drawing the forces as a chain and calculating the resultant using the x and y components.
cr7einstein

## Homework Statement

Hi all,

It is a homework problem, but I really don't quite understand the question. It reads-

"3 forces of magnitudes 10N, 20N, and 30N acting on a point are parallel to the sides of an equilateral triangle, taken in order. Find their resultant"

## The Attempt at a Solution

What I think is, as the resultant of 3 vectors forming a closed figure is 0, the answer should be 0.But the answer is given as $$10 \sqrt{3}$$. How?

Draw a picture

Compute x and y components of each vector

Add x components to get x component of sum

Add y components to get y component of sum

If needed, put resultant back into magnitude angle form with Pythagoras and atan

Thanks.
The problem is I don't know how to draw the diagram, and why should the resultant not be zero? It does form a closed figure, doesn't it?

Also, aren't there 3 vectors? SO, am I supposed to take one of them along the X axis?

The resultant would only be zero if all three forces had the same magnitude.

Draw F1 along the x axis.
Draw F2 with a 120 deg angle wrt the +x axis.
Draw F3 with a 210 deg angle wrt the +x axis.

+1 to that.

Note the question says the vectors are parallel to the sides of an equilateral triangle not that they are arranged in a triangle.

cr7einstein said:
What I think is, as the resultant of 3 vectors forming a closed figure is 0
You are confusing two concepts. If you take the forces acting at a point and draw them as a chain in which:
• The head end of one touches the tail of the next
• Each line is parallel to the force it represents
• The length of each line is proportional to the magnitude of the force
then the line from the tail of the first in the chain to the head of the last in the chain represents the resultant. In particular, if it forms a closed polygon then the resultant is zero.
But the equilateral triangle here does not represent the force magnitudes. It is only telling you the directions of the forces. If you try to draw a triangle in which all the angles are 60 degrees but you make the lines different lengths then it won't close.

## 1. What is the formula for calculating the resultant of 3 vectors along the sides of an equilateral triangle?

The formula for calculating the resultant of 3 vectors along the sides of an equilateral triangle is R = A + B + C, where R is the resultant vector and A, B, and C are the individual vectors along the sides of the triangle.

## 2. How do you determine the direction of the resultant vector?

The direction of the resultant vector can be determined by using the law of cosines and trigonometry. The angle between the resultant vector and the first vector can be calculated using the formula cos⁡θ = (A² + R² - B²) / (2AR), where θ is the angle and A and B are the magnitudes of the first and second vectors, respectively. The direction of the resultant vector will be in the same direction as the first vector.

## 3. Can the resultant of 3 vectors along the sides of an equilateral triangle be negative?

No, the resultant of 3 vectors along the sides of an equilateral triangle cannot be negative. Vectors are represented by both magnitude and direction, and the resultant vector will always have a positive magnitude and direction in the same direction as the first vector.

## 4. How does the angle between the vectors affect the magnitude of the resultant vector?

The angle between the vectors can affect the magnitude of the resultant vector. If the vectors are in the same direction, the magnitude of the resultant vector will be equal to the sum of the magnitudes of the individual vectors. However, if the vectors are not in the same direction, the magnitude of the resultant vector will be less than the sum of the individual vectors.

## 5. Can the resultant of 3 vectors along the sides of an equilateral triangle be greater than the sum of the individual vectors?

Yes, the resultant of 3 vectors along the sides of an equilateral triangle can be greater than the sum of the individual vectors. This can occur when the vectors are not in the same direction and the angle between them is less than 60 degrees. In this case, the magnitude of the resultant vector will be greater than the sum of the individual vectors.

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