In this explainer, we will learn how to find the resultant of two forces acting on one point and how to find the direction of the resultant.

We start by defining a force and exploring its properties.

### Definition: Force

Force is defined as the effect of one natural body on another. Each force is described in terms of its magnitude (size), direction, point of action, and line of action.

We often represent a force by using the notation .

### Properties: Forces

- The
*magnitude*of a force is its size, which is measured in newtons (N). By using a directed line segment to represent the force , and drawing the line to a suitable scale, we can use the length of the line to denote the magnitude, . - The
*direction*of a force is the direction in which it acts. By using a directed line segment to represent the force , we can use the direction of the arrow to show the direction of the force. - The
*point of action*of a force is the point at which it is applied. - The
*line of action*of a force is a geometric way to represent how the force is applied. It is drawn as a line through the point of action in the same direction as .

For example, the diagram below shows the force represented by the directed line segment .

The magnitude of the force is determined by . The direction of the arrow corresponds to the direction of . The point of action is . The line of action is indicated by extending in the same direction (as shown by the dotted line).

A force acting on a body is represented by vector . When two forces act on a body, we call their resultant the force that describes their combined effect.

### Definition: Resultant Force

When two forces, and , act on a body at the same point, the combined effect of these two forces is the same as the effect of a single force, called the resultant force.

The resultant force, , is given by

The vector equality can be represented in two ways, as illustrated in the following diagram.

As , , and are three sides of a triangle, we can use either the law of sines or the law of cosines in the triangle to find the resultant of the two forces, the angles between the resultant and the forces, or any other unknown.

Let be the angle between forces and , the angle between and , and the angle between and , as shown in the diagram below.

The law of sines in this triangle gives us where , , and are the magnitudes of , , and respectively.

As for all , we find the relationship given in the following box.

### Property: Law of Sines in a Triangle Formed by Two Forces and Their Resultant

We have where , , and are the magnitudes of , , and , respectively, is the angle between forces and , is the angle between and , and is the angle between and .

Applying the law of cosines in our triangle now, we find that

As for all , we find the relationship given in the following box.

### Property: Law of Cosines in a Triangle Formed by Two Forces and Their Resultant

We have where , , and are the magnitudes of , , and , respectively, and is the angle between forces and .

By taking the square root of both sides of the above equality and recalling that the magnitude of a vector is positive, we can obtain an explicit formula for , the magnitude of . It is also straightforward to derive an accompanying formula for the direction of . We state these results below.

### Formula: The Magnitude and Direction of the Resultant of Two Forces

Let be the resultant force of two forces, and , that act at a single point with an angle between them. Then, where , , and are the magnitudes of , , and , respectively, and is the angle between and .

Let us start with an example in which the magnitude of the resultant of two forces acting at a point is determined.

### Example 1: Finding the Magnitude of the Resultant of Two Forces

Two forces of magnitudes 35 N and 91 N are acting at a particle. Given that the resultant is perpendicular to the first force, find the magnitude of the resultant.

### Answer

It will be convenient to assume that the first force acts horizontally. Let us call this force and the other force . The resultant of these forces, , acts vertically as it is perpendicular to , as shown in the following figure.

The force can be represented by an arrow with its tail at the head of and its head at the head of , as shown in the following figure.

The resultant force is given by

As and are perpendicular, we see that the two forces and their resultant form a right triangle. Therefore, applying the Pythagorean theorem gives

It is worth noting that the Pythagorean theorem is just a special case of the law of cosines.

Substituting in the values of and , we find that

Note that as the magnitude of a vector is always positive, N is not a valid solution.

The magnitude of the resultant of the forces is 84 N.

Let us now look at an example in which the direction of the line of action of the resultant of two forces acting at a point is determined.

### Example 2: Finding the Direction of the Resultant of Two Forces Acting at the Same Point

Two perpendicular forces of magnitudes 88 N and 44 N act at a point. Their resultant makes an angle with the 88 N force. Find the value of .

### Answer

It will be convenient to assume that one of the forces acts horizontally. Let us call this force and the other force , as shown in the following figure.

By choosing to make correspond to the line adjacent to , we have chosen this force to be the 88-newton force. The magnitude of is 44 newtons; therefore, the magnitude of is half that of . The magnitude of the resultant of the forces, , can be expressed as

We can see from this that

Taking square roots, we have that

Applying the law of sines in the triangle gives

As , we have

We have, therefore, that

Let us now look at an example in which the magnitude and direction of the line of action of the resultant of two perpendicular forces are known and the magnitudes of the forces must be determined.

### Example 3: Finding Two Forces given the Magnitude and Direction of Their Resultant

Two perpendicular forces, and , act at a point. Their resultant, , has magnitude 188 N and makes an angle of with . Find the magnitudes of and .

### Answer

The perpendicular forces, and , and their resultant are shown in the following figure.

We see that and are perpendicular and the resultant makes an angle of with . As we have a right triangle, we have and

has a magnitude of 94 N, and has a magnitude of N.

Let us now look at an example involving two nonperpendicular forces.

### Example 4: Finding a Missing Force given Information About the Resultant Force

The angle between forces and is , and the measure of the angle between their resultant and is . If the magnitude of is 28 N, what is the magnitude of ?

### Answer

The following figure shows the forces and and their resultant . The forces act at a point .

The resultant forces and form a parallelogram whose diagonal through is the resultant.

The angle, , between and the resultant of and is given by

We can now add this angle and its alternate interior angle in our diagram as shown.

Applying the law of sines in the triangle formed by , , and , we find that that is,

The magnitude of is given as 28 N, so the magnitude of is also 28 N.

Let us look at our last example where the direction of one of the forces is reversed.

### Example 5: Finding the Magnitude of Two Forces of Identical Magnitude given Their Resultant at Two Cases

Two forces, both of magnitude N, act at the same point. The magnitude of their resultant is 90 N. When the direction of one of the forces is reversed, the magnitude of their resultant is 90 N. Determine the value of .

### Answer

Let us represent the first situation.

When we add two forces, and , the resultant is the diagonal of the parallelogram formed by and , with its tail being the point of application of and . If the two forces have the same magnitude, then the parallelogram is a rhombus, and the two forces and their resultant form an isosceles triangle, as shown in the following diagram.

Applying the law of cosines, we find that with , , and .

Since , we have

If we now reverse the direction of one of the forces (for symmetry reasons, it does not matter which force has its direction reversed; we will get the same result), the resultant will still be the diagonal of a rhombus congruent to the previous one, but it will be the other diagonal, and the angle between forces and will be .

The magnitude of is the same as the magnitude of , .

Applying the law of cosines in the triangle formed by , , and their resultant gives us that is,

We are told that the magnitude of the resultant is the same in both cases, 90 N. Hence, we have which means that

This is true only if , that is, if . Forces and are, thus, perpendicular.

Hence, we have

It is worth noting that, in the previous example, we could have concluded that the two forces are perpendicular with simple geometric considerations: the diagonals in a rhombus have the same length only if the rhombus is a square.

Let us now summarize what has been learned in these examples.

### Key Points

- Force is defined as the effect of one natural body on another. Each force is described in terms of its magnitude (size), direction, point of action, and line of action. We often represent a force by using the notation .
- The resultant, , of two forces, and , acting on a body at the same point is a single force that is given by
- The combined effect of and is the same as the effect of only .
- , , and are three sides of a triangle or two adjacent sides and a diagonal of a parallelogram.
- Applying the law of sines in the triangle formed by two forces and and their resultant, , gives where , , and are the magnitudes of , , and , respectively, is the angle between forces and , is the angle between and , and is the angle between and .
- Applying the law of cosines in the triangle formed by two forces and and their resultant, , gives where , , and are the magnitudes of , , and , respectively, and is the angle between forces and .
- Let be the resultant force of two forces, and , that act at a single point with an angle between them. Then, where , , and are the magnitudes of , , and , respectively, and is the angle between and .