Understanding vectors

The world of vectors surrounds our every move. Our world is one of pushes and pulls. Suppose you are leaning against a wall. What keeps you from sliding down the wall? Forces act against your body, including the pull of gravity. There are forces against your back, feet, shoulders, and head, if your head is touching the wall. Each of these forces has a direction in terms of an angle from the horizontal axis and amount of force exerted, called the magnitude. Such quantities that involve both direction and magnitude are called vectors.
Here's some examples of vectors. Suppose you are pulling a cart up a 20 degree incline, requiring an effort of 85 pounds. The magnitude is 85 pounds and the direction is 20 degrees.

As another example, suppose you are driving due west at 60 miles per hour. The magnitude is the speed, 60 miles per hour. The direction of the vector is due west.

Signs on a post that point in different directions and show the distance to each city are all vectors. Each sign defines a vector for each city.

Some quantities are described completely by given a magnitude, such as temperature. Suppose your body temperature is 98 degrees. Then the magnitude is 98, but there is no direction. These quantities are not vectors, they are scalars. Vectors can be multiplied by scalars.

A line segment that is given a direction is called a directed line segment. Geometrically, a directed line segment in a vector. Suppose we have line segment PQ. P is the initial point and Q is the terminal point. Vectors are often denoted by a boldface letter, such a v.

Suppose we have two vectors, v and w. There are four possible relationships between the two vectors. If v and w have the same magnitude and direction, they are equal. They can have the same magnitude and different direction. They can have the same direction and different magnitude, or they can have different magnitude and different direction.

To show that two vectors are equal, we take the initial and terminal points of both vectors and use the distance formula to get the magnitude. For example, suppose v has a starting point (0,0) and terminal point (4, 5) and w has a starting point of (-1, -2) and terminal point (3, 3). Using the distance formula learned in geometry, you see that magnitude of v = square root of 41 and the magnitude of w = square root of 41.
One way to show that v and w have the same direction is to get the slope of the lines on which they lie. Using the slope formula, we see both are 4/5 and both are pointed upward and to the right.

A vector can be multiplied by a real number. Suppose a vector is multiplied by 3. The vector will be 3 times as long as the original vector pointing in the same direction. If the vector is multiplied by a negative number, it points in the opposite direction from the original.

There are many other concepts dealing with vectors, such as vectors in the rectangular coordinate system, representing vectors in rectangular coordinates and finding its magnitude and operations of vectors. Those are topics for a more detailed discussion on vectors.

This introduction to vectors is one I use when working with students who are having difficulty understanding the basics of vectors.