How To Find The Magnitude Of A 3d Vector

In this explainer, nosotros will learn how to find the magnitude of a position vector in space.

There are a number of ways in which a 3D vector tin exist expressed, however, we are going to focus on component form and unit vector form.

Definition: Component and Unit Vector Representations

Consider the following indicate in 3D space.

The coordinates of betoken are , and describes the vector from the origin to point .

This vector tin can be represented in 2 ways:

Notation that these are but different forms of notation. The last grade uses the unit vectors , , and , which are vectors of magnitude ane in the -, -, and -directions respectively:

We should exist familiar with the fact that a unit of measurement vector has a magnitude (or length) of 1. We should also be familiar with finding the magnitude of a vector in 2nd by leveraging the Pythagorean theorem stated below:

Note that this is essentially the same problem as finding the magnitude of a vector in 2d. Recall that, given some vector , its magnitude is often represented using the notation or sometimes :

Mayhap a lesser known fact about the Pythagorean theorem is that it can be generalized up to as many dimensions as we require. We tin can leverage this fact to extend our 2D method to cover 3D.

Instead of finding the long diagonal of a rectangle, equally shown in our 2D example, we can use the Pythagorean theorem in 3D to find the long diagonal of a cuboid:

Ane might initially assume that since the 2-dimensional theorem uses squares (or square roots), so the iii-dimensional theorem would require cubes (or cube roots). Fortunately, this is not the case!

For this explainer, nosotros volition not exist going into the proof of the theorem, simply it is enough to recognize that finding the length of the long diagonal of a cuboid is essentially the same problem as finding the magnitude of a vector in 3D.

This is the fundamental insight we demand for our definition.

Definition: The Magnitude of a Vector in 3D

Consider vector in 3D space, where the vector can be expressed as:

The magnitude of is given by

Notation that it is common to but use the variables , , and in magnitude calculations:

In our magnitude adding, each component of the vector is squared before the sum is square rooted. This means nosotros will e'er end up taking the square root of a positive number, which is well defined but has ii solutions. For example, may be v or .

Information technology is worth noting that the magnitude of a vector is defined to be nonnegative. This means that we can safely ignore the negative solution for all magnitudes in this explainer.

Now that nosotros understand how to find the magnitude of a general vector in 3D, permit's practice applying this cognition in some examples.

Case 1: Finding the Magnitude of a 3D Vector

If , find .

Respond

This question gives u.s.a. vector in component form. Since the vector has iii components, nosotros recognize that it exists in 3D infinite.

An alternative mode to express using unit of measurement vectors is

Nosotros take been asked to find , which represents the magnitude (or length) of the vector. In gild to solve this question, we recollect that the magnitude of a vector in 3D space is given by where , , and stand for the components of the vector in the respective cardinal directions.

Our vector has the post-obit components:

To find its magnitude, we substitute these values into the formula:

With a small amount of simplification, we reach our reply.

With this type of question, it may occasionally be helpful to find a decimal approximation for your answer (perhaps if needed for comparisons). In this case, it is non necessary and since there is no way to helpfully reduce , we can leave our answer expressed as a radical.

Instance two: Finding the Magnitude of a 3D Vector Expressed in terms of Unit Vectors

If , detect .

Reply

This question takes a very similar form to our previous instance; withal, this time we are working with a 3D vector, , which has been given in terms of unit vectors.

Again, we have been asked to find the magnitude of this vector, then we can use the formula for the magnitude of a vector in 3D:

Our vector has the post-obit components:

It is worth noting that fifty-fifty though the unit of measurement vector does non announced to have a coefficient, the fact that information technology is present in our vector suggests otherwise. We should always be careful non to ignore coefficients of i, or in this case , when working with vectors expressed equally the sum of unit vectors. Over again, nosotros substitute our values into the formula and simplify to find :

Much similar our previous example, it is perfectly fine to leave our magnitude in the form of a radical.

The formula that we accept been using non only helps u.s. to observe the magnitude of a vector, but too tin exist used to find a missing component of a vector if we are given the magnitude.

Let's accept a look at an example of this.

Example 3: Finding the Value of an Unknown Component of a Vector Using Its Magnitude

If and , find all the possible values of .

Respond

For this example, nosotros have been given the magnitude of a 3D vector and we must utilize this information to find an unknown component. The coefficient for the unit vector is given by the parameter . This is the unknown that nosotros must work toward finding.

Our vector has the post-obit components:

We can substitute these values into the formula for the magnitude of a vector in 3D:

In this case, we have some other piece of information. The magnitude of the vector . In order to solve for our unknown, we will need to substitute this into our equation also:

To go on, we can foursquare both sides then simplify:

At this phase, we have an equation for . Here, we can have the foursquare root of both sides of the equation:

We should remember that taking the square root of a number has both a positive and a negative solution. Since we are finding the coefficient of a vector (and not a magnitude), nosotros cannot ignore the negative solution.

Our answer is that the possible values for are 2 and .

As a footnote to the previous example, consider that when given the magnitude of a vector, we are only able to find the value of a single unknown component of that aforementioned vector. Had at that place been more i unknown component, our equation might accept looked something like this:

Since the above equation has 2 unknowns, in that location would be an infinite number of solutions, and so we would non be able to find a unique pair of values for and .

Moving into to our adjacent example, we recollect that when working with vectors, operations such equally addition and subtraction are important tools. In order to find the magnitude of a vector, we may first need to use these operations to find its components.

Instance 4: Solving Problems Involving the Magnitude of a Vector

Given that and , make up one's mind .

Reply

Retrieve that when adding 2 vectors, the individual components simply add together. The given expression is itself a vector, and we have been given its components, along with the components of the alone vector . Since the components of are unknown, we volition stand for them with using , , and :

This allows u.s. to form the equation

In gild to isolate our unknowns, we can subtract vector , or , from both sides of this equation, leaving usa with just vector , or , on the right-hand side:

At this point, we could set up iii separate equations for the , , and directions; however, this would be unnecessary. Recalling the properties of vector addition and subtraction, nosotros can simplify the left-hand side directly by treating each component separately:

Since we defined to stand for vector , we have found that

Now that we have found the components of vector , we can find its magnitude:

We have now completed the question. Since there are no useful simplifications to perform, we can leave our respond in surd form.

Notation that the properties of vector addition and subtraction can be used when dealing with systems involving coordinate points.

Consider a system with 2 points and . Imagine nosotros wanted to detect the altitude between these 2 points.

This problem is the same as finding the magnitude of the vector between points and , in other words, .

We know that

Vector can easily exist plant by recalling that

Nosotros could and so find the altitude betwixt the ii points by applying our magnitude formula:

I very important distinction we will run across is that the magnitude of a sum or difference is not necessarily the aforementioned as the sum or departure of two magnitudes:

We volition meet why this is the case in the next example.

Example five: Solving Problems Involving the Magnitude of a Vector and Vector Addition

If and , decide and .

Answer

The question has given us two 3D vectors and asked us to find the magnitude of the sum of the vectors, , forth with the sum of the magnitude of the vectors, . While these ii things may look very similar, we should exist careful non to assume they are equivalent.

Permit's start with . We will first demand to find the components of the vector by calculation our 2 given vectors:

With some simplification, we discover our components. The coefficients for each of our unit vectors can now be substituted into the formula for the magnitude of a vector:

Now what almost ? At this indicate, finding the magnitude of a known vector should be familiar. We start by finding :

Then, we observe :

Putting these ii pieces of data together gives us our answer:

For the sake of comparison, we may choose to observe a decimal approximation for these two quantities:

Certainly, in this case, we have shown that .

If we choose to examine and more than closely, we may realize that this is essentially the triangle inequality, equally shown below!

The magnitude of the sum of 2 vectors can never exist greater than the sum of the magnitudes:

Note that in that location is a unmarried example when these quantities are equal, which occurs when the vectors and betoken in the same direction. In other words, and are parallel.

This logic can besides be applied to compare with . Doing so leads to a similar consequence, which is summarized in the fundamental points beneath.

Key Points

• The magnitude of a vector represents its length and is defined to always exist a positive number.
• The term represents the magnitude of vector .
• Given that , its magnitude can be found using the following formula:
• If ii vectors and have the same direction, then
• If two vectors and exercise non have the same direction, then

Source: https://www.nagwa.com/en/explainers/636174202405/

Posted by: yockeybegry1954.blogspot.com

Share this post