VQ - Define Vectors Lesson

Define Vectors

In previous courses, you have dealt with many scalar quantities, like speed. You can tell me a car is traveling 45 miles per hour and that is an example of a scalar quantity. In this module, we will explore vectors. A vector is a quantity that has both magnitude and direction – so if I say a car if traveling 45 mph due west, then that is a vector!

Here are some more examples of scalar and vector quantities so that you can see the difference.

Scalar Quantities	Vector Quantities
a boat is traveling at 15 mph	a boat is traveling at 15 mph in the direction of 25°
a hiker walks 30 paces	a hiker walks 30 paces due north

A vector is drawn geometrically by a directed line segment. The vector shown has an initial point, A, the point where the vector begins is also called the tail. The point where the vector ends, B, is called the terminal point or the tip. We could denote this vector equation image indicator $LaTeX: \overline{AB},\:\vec{a},\:or\:a$ $\overline{AB},\:\vec{a},\:or\:a$ .

If the initial point of a vector is at the origin, then we say the vector is in standard position. The direction of the vector is the angle between the vector and the positive x-axis, in the image below the direction is 35°. The length of the line segment represents the magnitude of the vector.

vector AB with segment a at 35° on graph

So imagine that I threw a ball 20 feet per second due North, that vector would look like the one below.

ray on y-axis denoted at 20

Recall that when we discuss navigational direction, we consider the angle to be measured clockwise from due north. For instance, if I want to travel in the direction of 150° my angle would look like this:

compass circle with 150° direction

Match the statements below with the appropriate visual representation of the vector (estimate the length).

Important Fact: If I want to represent a vector with more magnitude, I draw a longer vector.

As you study vectors, it will be important to take some properties of lines that you know, and apply them.

Vectors are parallel if they have the same or opposite direction but not necessarily the same magnitude. In the figure, all of the vectors are parallel except $LaTeX: \vec{b}$ $\vec{b}$ , so all the other vectors are parallel, denoted by: a||c||d||e||f.
Vectors are equivalent if they have the same magnitude and direction. In the figure, a = c because they have the same magnitude and direction. Notice that vectors d and e have the same magnitude but opposite directions so they are not equivalent.
Opposite vectors are those that have the same magnitude, but opposite directions. The vector opposite d is written –d, in the figure –d = e.

multiple parallel vectors, vector b is not parallel to the others

Component Form of a Vector

The component form of a vector (also called wedge form) makes the horizontal and vertical parts of a vector easy to see!

vector a <4, 3>

We can see in the image above that vector a has a horizontal component of 4 and a vertical component of 3. Recall that the magnitude of the vector is the length of the vector, how might you find the magnitude of the vector?

equation image indicator $LaTeX: \rVert{a}\lVert=\sqrt[]{\left(4\right)^2+\left(3\right)^2}=\sqrt[]{16+9}=\sqrt[]{25}=5$ $\rVert{a}\lVert=\sqrt[]{\left(4\right)^2+\left(3\right)^2}=\sqrt[]{16+9}=\sqrt[]{25}=5$

So in general, the magnitude of a vector in component form is:

equation image indicator $LaTeX: a=\left(x_1,\:y_2\right)\\ \lVert{a}\rVert=\sqrt[]{\left(x_1\right)^2+\left(y_1\right)^2}$ $a=\left(x_1,\:y_2\right)\\ \lVert{a}\rVert=\sqrt[]{\left(x_1\right)^2+\left(y_1\right)^2}$

Sometimes, you'll be given a vector that is not in standard position. How might you determine the components of that vector?

IMAGES CREATED BY GAVS