# How to find the magnitude of the resultant of three vectors

So let's suppose that we call it South. If she started south, then upon making a right-hand-turn, she is now heading west. And after making a second right-hand-turn, she is heading North. The diagram at the right depicts the physical situation. The Pythagorean theorem can be used to determine the magnitude of the resultant. The work is shown below.

Chapter 2 - Force Vectors

They are shown in the graphic at the right. The resultant is the hypotenuse of a right triangle that has lengths of The results of this process are organized in the table below. Results are entered to a couple of decimal places. Correction for significant digits and rounding are made at the end of the problem.

Kinematics Newton's Laws Vectors and Projectiles. What Can Teachers Do Subscription Selection. Student Extras. We Would Like to Suggest Sometimes it isn't enough to just read about it. You have to interact with it! And that's exactly what you do when you use one of The Physics Classroom's Interactives. All three Interactives can be found in the Physics Interactive section of our website and provide an interactive experience with the skill of adding vectors.

The walk from A to E is 12 squares to the right East. Since each square is 20 meters wide, this would be meters. There is no north-south component to this leg of the two-leg walk.

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Close Menu. JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. Minimum and maximum resultant of three vectors. Thread starter Vatsal Goyal Start date Feb 2, Tags maximum minimum resultant resultant vector vector.

Vatsal Goyal. Homework Statement In order to solve a question, I need to find the minimum and maximum resultant of three vectors, their magnitudes are given to me.

But I am not sure about the minimum part. As it cannot form a triangle, it can't be zero. Am I correct? If I am, then is there a strong reason for my answer to be true.

Homework Helper. Insights Author. Lets start with like you said. Now let's start with the 1 vector starting at the end of that resultant, and being able to spin at any angle. What is the magnitude of the resultant. Perhaps express it as a function of the angle, and see where the minimum occurs.In the introduction to vectorswe discussed vectors without reference to any coordinate system.

By working with just the geometric definition of the magnitude and direction of vectors, we were able to define operations such as addition, subtraction, and multiplication by scalars. We also discussed the properties of these operation. Often a coordinate system is helpful because it can be easier to manipulate the coordinates of a vector rather than manipulating its magnitude and direction directly. When we express a vector in a coordinate system, we identify a vector with a list of numbers, called coordinates or components, that specify the geometry of the vector in terms of the coordinate system.

Here we will discuss the standard Cartesian coordinate systems in the plane and in three-dimensional space.

Using the Pythagorean Theorem, we can obtain an expression for the magnitude of a vector in terms of its components. Can you calculate the coordinates and the length of this vector? To find the coordinates, translate the line segment one unit left and two units down.

The below applet, repeated from the vector introductionallows you to explore the relationship between a vector's components and its magnitude. The magnitude and direction of a vector. The two defining properties of a vector, magnitude and direction, are illustrated by a red bar and a green arrow, respectively.

More information about applet. The vector operations we defined in the vector introduction are easy to express in terms of these coordinates. The below applet, also repeated from the vector introductionallows you to explore the relationship between the geometric definition of vector addition and the summation of vector components. The sum of two vectors. You may have noticed that we use the same notation to denote a point and to denote a vector. We don't tend to emphasize any distinction between a point and a vector.

You can think of a point as being represented by a vector whose tail is fixed at the origin. You'll have to figure out by context whether or not we are thinking of a vector as having its tail fixed at the origin. A unit vector is a vector whose length is one. Here is one way to picture these axes. Stand near the corner of a room and look down at the point where the walls meet the floor.

The negative part of each axis is on the opposite side of the origin, where the axes intersect. Three-dimensional Cartesian coordinate axes. A representation of the three axes of the three-dimensional Cartesian coordinate system. The origin is the intersection of all the axes. The branch of each axis on the opposite side of the origin the unlabeled side is the negative part. You can drag the figure with the mouse to rotate it.

If you do that, you will be living in a mathematical universe in which some formulas will differ by a minus sign from the formula in the universe we are using here. Your universe will be just as valid as ours, but there will be lots of confusion. We suggest you live in our universe while studying from these pages. Sums, differences, and scalar multiples of three-dimensional vectors are all performed on each component. A vector in three-dimensional space.

You can drag the head of the green arrow with your mouse to change the vector. You can also drag the heads of the purple vectors to change just one of the coordinates of the vector. The standard unit vectors in three dimensions. Moving them with the mouse doesn't change the vectors, as they always point toward the positive direction of their respective axis.A variety of mathematical operations can be performed with and upon vectors.

One such operation is the addition of vectors. Two vectors can be added together to determine the result or resultant. This process of adding two or more vectors has already been discussed in an earlier unit. Recall in our discussion of Newton's laws of motion, that the net force experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object. That is the net force was the result or resultant of adding up all the force vectors.

During that unit, the rules for summing vectors such as force vectors were kept relatively simple. Observe the following summations of two force vectors:. These rules for summing vectors were applied to free-body diagrams in order to determine the net force i. Sample applications are shown in the diagram below. In this unit, the task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions.

For example, a vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below. There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are:.

The Pythagorean theorem is a useful method for determining the result of adding two and only two vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

To see how the method works, consider the following problem:. This problem asks to determine the result of adding two displacement vectors that are at right angles to each other.

The result or resultant of walking 11 km north and 11 km east is a vector directed northeast as shown in the diagram to the right. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant i. The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of Laterthe method of determining the direction of the vector will be discussed.

Let's test your understanding with the following two practice problems. In each case, use the Pythagorean theorem to determine the magnitude of the vector sum. When finished, click the button to view the answer. The direction of a resultant vector can often be determined by use of trigonometric functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle.

The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The tangent function relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The three equations below summarize these three functions in equation form.

These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement. The process begins by the selection of one of the two angles other than the right angle of the triangle.

Once the angle is selected, any of the three functions can be used to find the measure of the angle.

Write the function and proceed with the proper algebraic steps to solve for the measure of the angle. The work is shown below. Once the measure of the angle is determined, the direction of the vector can be found. In this case the vector makes an angle of 45 degrees with due East.Last Updated: September 14, References Approved. To create this article, 15 people, some anonymous, worked to edit and improve it over time.

This article has been viewedtimes. Learn more A vector is a geometrical object that has both a magnitude and direction. Calculating the magnitude of a vector is simple with a few easy steps. Other important vector operations include adding and subtracting vectorsfinding the angle between two vectorsand finding the cross product.

To find the magnitude of a vector, first determine its horizontal and vertical components on their respective number lines around the origin. Next, draw the horizontal and vertical components to plot the point where they intersect. Then draw a line from the origin to that point, creating a vector triangle, which is a right triangle. For more information on finding the magnitude of a vector, including using a modified formula when the vector is away from the origin, scroll down!

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Method 1 of Determine the components of the vector. Every vector can be numerically represented in the Cartesian coordinate system with a horizontal x-axis and vertical y-axis component. Draw a vector triangle. When you draw the horizontal and vertical components, you end up with a right triangle. The magnitude of the vector is equal to the hypotenuse of the triangle so you can use the Pythagorean theorem to calculate it.

### Minimum and maximum resultant of three vectors.

Rearrange the Pythagorean theorem to calculate the magnitude. Solve for the magnitude. Using the equation above, you can plug in the numbers of the ordered pair of the vector to solve for the magnitude. Vector magnitudes can be decimals. Method 2 of Determine the components of both points of the vector.

If you are given a vector that is placed away from the origin of the Cartesian coordinate system, you must define the components of both points of the vector.

For example, the vector AB has an ordered pair for point A and point B. Use a modified formula to solve for the magnitude. Plug in the numbers of your ordered pairs and calculate the magnitude.In this article, you will learn what the resultant force also known as net force is, and how to find it when an object is subject to parallel forces as well as non-parallel forces with the help of examples. When an object is subject to several forces, the resultant force is the force that alone produces the same acceleration as all those forces.

The reason why the resultant force is useful is that it allows us to think about several forces as though they were a single force.

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This means that to determine the effect that several forces have on an object, we only need to determine the effect that a single force has. If we know the mass m of an object and the acceleration a produced by the forces that act on it, we can find the resultant force using Newton's Second Law. Indeed, according to Newton's Second Law, the force F that alone produces the acceleration a on an object of mass m is:.

Which indicates that the resultant force R has the same direction as aand has magnitude equal to the product m a. For example, if a box of 1. Often, however, we know the forces that act on an object and we need to find the resultant force. Experiments show that when an object is subject to several forces, F 1F 2Notice that this is not a mere sum of the magnitudes of the forces, but the sum of the forces taken as vectorswhich is more involved because vectors have both a magnitude and a direction that we need to consider when doing the sum.

According to the above equation, if an object is subject to no forces, then the resultant force is zeroand if an object is subject to only one force, then the resultant force is equal to that force.

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These two cases are pretty simple, but what about an object subject to two or more forces? How do we perform the vector sum then? To explain this clearly, we will now go through all the cases that can happen, from simple ones in which all the forces are parallel, to more complex ones in which the forces are not parallel, and show how to find the resultant force in each of them with the help of examples. Let's start with the simple case in which an object is subject to two forces that act in the same direction:.

The resultant force is in the same direction as the two forcesand has the magnitude equal to the sum of the two magnitudes :. Let's consider the case in which an object is subject to two forces that act in opposite directions. The resultant force will be zero because two opposite forces cancel each other out. To find the resultant force in this case, we first sum all the forces that go in one direction, and then all the forces that go in the other direction:.

In the previous cases, we have forces that are all parallel to one another. It's time to consider the case in which an object is subject to two forces that are not parallel. For example, let's assume that we have a block subject to two forces, F 1 and F 2. Since one of the two forces is horizontal, for convenience, we choose the x -axis horizontal, and the y -axis vertical, and we place the origin at the center of our block:.

The next step is to determine the x and y components of all the forces that act on the block :. If we sum all the x components, we will get the x component of the resultant force :.

Similarly, if we sum all the y components, we will get the y component of the resultant force :. At this point, we know the x and y components of Rwhich we can use to find the magnitude and direction of R :. The magnitude of R can be calculated by applying Pythagoras' Theorem :.

Finally, let's examine the case in which an object is subject to more than two non-parallel forces. For example, suppose we have an object that is subject to three forces, F 1F 2and F 3.

We can find the resultant force R using the same process that we used in the previous case of two non-parallel forces. Then, we determine the x and y components of the individual forces:. Again, the x component of the resultant force R is the sum of all x components:. Similarly, the y component of R is the sum of all y components:. Finally, let's calculate the magnitude and direction of R using its two components R x and R y :. To express the direction of Rwe need to calculate the direction angle i.

The process that we used in this case and in the previous one to find the resultant force when the forces are not parallel can also be used when all the forces are parallel.

## Math Insight

For a better experience, please enable JavaScript in your browser before proceeding. Finding the direction of the resultant vector of three vectors?

Hi, I was wondering how one would go about finding the direction of the resultant of three vectors, when performing a vector addition of three vectors. I mean I know how to find the magnitude, by summing all the squares of the component vectors, i and j, and square rooting that sum, but how does one get the direction of that third vector?

The whole thing forms a polygon, for god's sake. Have you heard of the "Tip to tail method"? For this specific problem I have a motorist drives south at Split it into your x and y components.

Use your trigonometric funtions to find the direction and use the pythagorean theorem to find the magnitude. These are three main vectors I computed. I already found the magnitude. I just can't find the direction now of these three.

The components of the NW vector you mean? Tail to tip method says -nothing- about components. Point is, now if i use the tail-tip method, I have a polygon. So, you do want to do it graphically? You can do it both ways.

If you are instructed to do it algebraically, do it the way I said. If you are instructed to do it graphically, use tip to tail.

If given no specific method, use whichever is more comfortable for you. It doesn't matter if you have a polygon; You don't have to make a triangle with the vectors.