Vectors in Two & Three Dimensions

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  • 0:04 What Is a Vector?
  • 0:38 Two-Dimensional Vectors
  • 1:24 Three-Dimensional Vectors
  • 2:40 Adding & Subtracting…
  • 3:12 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Two-dimensional and three-dimensional vectors are very similar, and operations that can be performed on two-dimensional vectors can also be performed in similar ways on three-dimensional vectors. In this lesson, learn about both types of vectors.

What Is a Vector?

Zadie lives in Chicago and decides to go on vacation, so she gets on a plane and flies 200 miles. Where did she end up?

You can't know where she ends up without first knowing what direction she flew in, right? This is an example of a vector, which is a quantity that has both a magnitude and a direction. In order to know where Zadie went on her vacation, you need to know both how far she went and in what direction.

Many other physical quantities, such as force, velocity, and momentum, are also vectors. Vectors may be either two dimensional or three dimensional, depending on the situation.

Two-Dimensional Vectors

One way to represent a two-dimensional vector is with vector components, which simply tell you how far the vector goes in each direction. For example, a vector with an x-component of 4 and a y-component of 3 that started at the origin would end at coordinates (4,3).


2D vector components


The magnitude of a vector is the total amount of the quantity represented by the vector. For a two-dimensional vector, the magnitude is equal to the length of the hypotenuse of a triangle in which the sides are the x- and y-components. Therefore, if you know the two components of the vector and want to find the magnitude, you can use the Pythagorean Theorem. You can also use the tangent function to find the angle that the vector makes with the x-axis. For the vector shown here, the magnitude would be 5, and the angle it makes with the x-axis would be 37 degrees.


magnitude and direction of a 2D vector


Three-Dimensional Vectors

Three-dimensional vectors are just like two-dimensional vectors, but there is just one more direction to keep track of. Just like two-dimensional vectors, you can represent a three-dimensional vector using the three components in each of the three directions. There are few different ways that you might see a three-dimensional vector written. Sometimes, you may see the three components written inside brackets like this: < >

Other times, the components of a vector may be designated using the unit vectors: i, j, and k, which simply show which direction each component is in: i represents the x-component of the vector, j represents the y-component, and k represents the z-component. You may see this notation for two-dimensional vectors, but it's even more common for three-dimensional vectors.

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