Vectors

The concept of a vector is used extensively throughout science, mathematics, and engineering. We can use vectors for tasks like describing the trajectory of an airplane. In subjects such as pre-calculus and physics, we will need a strong grasp of vectors. Before starting, we may want to look at some supplementary materials regarding vectors. Closely related topics include the equation of a plane, cross products, and adding and subtracting vectors. These topics will help us develop a stronger understanding of how vectors work.

What is a vector?

A vector is a quantity that has both magnitude and direction. To make this concept more concrete, imagine a train that is traveling at 70 miles per hour. In many real-world contexts, this simply isn't enough information. If we're managing a system of trains, we would need to know both their speed and their directions. If we knew that the train is traveling northwest at 70 miles per hour, then we could create a vector that includes both pieces of information.

All sorts of physical phenomena are vectors, including acceleration, force, and velocity. This is why they're so prominent in physics. A vector begins at an initial point of A and ends at a terminal point of B.

We can call this "vector AB." When working with vectors the common notation is to place an arrow over the A and the B, $\overrightarrow{\text{AB}}$ .

When we place a vector on a graph, we can see that it has two components, one vertical and one horizontal.

We can call the component form of this vector $\left(x,y\right)$ .

Notice that this situation looks very similar to other math problems involving triangles. Indeed, if we know the values of each component, we can use the Pythagorean Theorem to calculate the magnitude of the vector. For instance, if we knew that $x=4$ and $y=3$ , we could calculate the hypotenuse. Recall that the Pythagorean Theorem states that the lengths of the legs and the hypotenuse of a triangle are connected by the equation

$c_2=a^2+b^2$ .

We can insert these values in for a and b, which represent the legs of the triangle.

$c^2=4^2+3^2$

$c^2=16+9$

$c^2=25$

Now we take the square root of both sides.

$\sqrt{c_2}=\sqrt{25}$

$c=5$

Vector practice problems

1. A pitcher throws a ball at 55 miles per hour. Can a vector represent this?

A vector has both a magnitude and a direction. Since no direction is given to us, this is not a vector.

2. A helicopter is traveling southeast at 120 miles per hour. Can we use a vector to represent this?

Since we are given both a speed and a direction, this can be represented by a vector.

3. If a vector has an x-component of 2 and a y-component of 3, what is the magnitude of the vector?

We'll use the Pythagorean Theorem to solve this.

$c_2=a^2+b^2$

We'll insert the x and the y-values in a and b.

$c^2=2^2+3^2$

$c^2=13$

$\sqrt{c^2}=\sqrt{13}$

$c=\sqrt{13}$

This is the value of our magnitude.

4. If a vector has an x-component of 0 and a y-component of 9, what is the magnitude of the vector?

We start with the Pythagorean Theorem.

$c_2=a^2+b^2$

We'll plug everything in.

$c^2=0^2+9^2$

$c^2=81$

$\sqrt{c^2}=\sqrt{81}$

$c=9$

Topics related to the Vectors

Cross Products

Adding and Subtracting Vectors

Equation of a Plane

Flashcards covering the Vectors

Precalculus Flashcards

CLEP Precalculus Flashcards

Practice tests covering the Vectors

Precalculus Diagnostic Tests

How to get further assistance with vectors

Vectors can be a perplexing topic to many students, but assistance is available. If your student is having trouble with vectors, a tutor will explain the concept and clarify any points that they don't understand. One helpful aspect of tutoring is that a tutor can develop a series of practice problems and use them to test students' understanding of the material. If you contact an Educational Director at Varsity Tutors, they will listen to your concerns and learn about the areas of math that your student wishes to improve upon. They will use this information to select a tutor who fits their personal needs. A tutor can operate off a personalized learning plan designed to help your student make progress toward their academic goals. Working alongside a tutor is ideal for high school students who may want extra assistance with their math classes. Contact Varsity Tutors as soon as you're ready to get started.