VM - 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.

Examples of Scalar and Vector Quantities
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 $LaTeX: \overline{AB},\:\vec{a},\:or\:a$ $\overline{AB},\:\vec{a},\:or\:a$ .

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 Compass Rose pointing North

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 Compass Navigational direction

Here are some other ways to apply direction:

Turning from N to S is 180 $LaTeX: ^\circ$ $^\circ$ clockwise or 180 $LaTeX: ^\circ$ $^\circ$ counterclockwise (sometimes called anticlockwise).
Turning from NE to SE is 90 $LaTeX: ^\circ$ $^\circ$ clockwise or 270 $LaTeX: ^\circ$ $^\circ$ counterclockwise.
Turning clockwise from NE to E is 45 $LaTeX: ^\circ$ $^\circ$ or 315 $LaTeX: ^\circ$ $^\circ$ counterclockwise.

Cardinal Direction:

cardinal direction example

Other ways to specify 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?

You can use the pythagorean theorem which says when you square the 2 legs of a right triangle and add them together, you get the square of the hypotenuse (length that is opposite of the right angle.)

Pythagorean Theorem

As you can see from the component vector, we can make a right triangle using 4 as the length of one leg and 3 as the length of the other leg.

Vector Component Form Triangle

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

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

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

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

Applications:

Vector Applications

What is the direction of the vector shown here?

Direction Example

Answer:

If you look at where the angle is placed in this diagram, you'll probably agree that we are measuring an angle of 30° away from the North.

In fact, the 30° angle is moving towards where East is.

In the physics style of giving a direction, we would write [N30°E], which is read as "North 30° East". The reference line (North) is given first, and then the number of degree away from it (30°) going towards another of the reference lines (East). Since all the way from North to East is a full 90°, we know that this could also be drawn showing that the vector is 60° (we get it from 90° - 30°) counterclockwise from the East. This would mean that we could also measure this vector as [E60°N] . Either one is still correct.

[CC BY-NC-SA 4.0] UNLESS OTHERWISE NOTED | IMAGES: LICENSED AND USED ACCORDING TO TERMS OF SUBSCRIPTION - INTENDED ONLY FOR USE WITHIN LESSON.