Unit vectors point in the direction of a vector and a normal vector is perpendicular to a vector. In this lesson, we will investigate how to determine unit and normal vectors.
Parallel and Perpendicular to a Path
Imagine you are walking along a straight line. Maybe it is along a sidewalk or along the straight section of a track around a high school football field. If you walk straight along the path, your direction can be considered a unit vector and its magnitude is 1. Let's say you are on the sidewalk and you decide to cross the street, so you turn off the sidewalk and walk perpendicular to it. Your new direction can be considered a normal vector. Let's investigate these two types of vectors and see how to calculate them.
Let's say we have a vector v notated as v = <a, b, c>. This means the vector is a-units in the x-direction, b-units in the y-direction and c-units in the z-direction. A unit vector is a vector that points in the direction of vector v but has a magnitude of 1 unit. Diagram 1 shows a vector and its unit vector.
Diagram 1: The red arrow is the given vector and the purple arrow is the unit vector
To determine the unit vector, we can use the notation:
The ∧ symbol over the u indicates that u is the unit vector. The denominator of each term is the magnitude of the vector v. To determine the magnitude of the vector, we use the Pythagorean theorem. The Pythagorean theorem is:
Let's work an example to see how to determine a unit vector.
Given: Vector v = <-1, 3, -4>. Determine its unit vector.
Solution: First, let's determine the magnitude of vector v.
Now, all we have to do is to plug the values into the original unit-vector expression.
It's time to move on to normal vectors.
Normal vectors are vectors that are perpendicular to another vector. Let's look at Diagram 2, which shows our original vector v and a couple of vectors normal to it.
Diagram 2: The red arrow is a given vector v, and the three purple vectors are normal vectors to v
Technically, there is an infinite number of normal vectors to any vector because the only criteria for a normal vector is that is 90° to the original vector. This is where the dot product comes in. The dot product between two vectors is:
The arrow over a letter indicates that it is a vector. It has a magnitude and a direction. Since the angle between a given vector and any normal vector is 90°, the right side of this equation is 0 because the cosine of 90° is 0.
The dot product between a given vector and the normal vector equals zero
The dot product can be calculated by multiplying the x-components of the given vector and normal vector, the y-components of the given vector and the normal vector and the z-components of the given vector and the normal vector. Then we sum the results.
Evaluating the dot product between two vectors
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We'll call vector b our normal vector, and since there are infinite normal vectors to a given vector, we can make up values for the x and y-components of the normal vector and calculate the z-component. This is best explained with an example, so let's do one!
Given: Vector v is v = <-1, 4, 6>. Determine a normal vector to vector v.
Solution: Setting up the dot product between the given vector v and its normal vector b, we get:
We will now make up values for the normal vector's x and y-components. Let's pick 3 for x and 4 for y.
This gives us the normal vector as:
A plot of the given vector and the normal vector shows they are 90° to each other.
The given vector is in blue and the normal vector is in purple
A unit vector is a vector that points in the direction of any vector but has a magnitude of 1 unit. Unit vectors have the ∧ symbol over them. The expression for a unit vector is:
Unit vector equation
The denominator is the magnitude of the given vector, which can be determined with the Pythagorean theorem.
A normal vector is a vector perpendicular to another vector. These vectors are 90° to each other. To determine a normal vector, we set the dot product between the vectors equal to zero. The equation for the dot product is:
Vector b in the equation is the normal vector to vector a. We can pick any two components of vector b and calculate the third component. This gives us the normal vector.
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