By Steven Holzner

A vector is a combination of exactly two values: a magnitude (like the speed of an object in motion) and a direction (such as the direction of an object in motion). All kinds of things can be described with vectors, including velocity, acceleration, displacement, magnetic fields, electric fields, and many more.

Vectors are defined by a magnitude (the length of the vector) and a direction. For example, take a look at the vector shown.

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In physics, vectors are written in bold type. The vector shown — called A — represents the displacement of a golf ball from the tee. Its length is 100 yards, and its direction is 15 degrees north of due east. That’s all you need to have a vector — a magnitude and a direction.

Now take a look at the two vectors shown here, A and B. These two vectors are considered equal, which is written as A = B.

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Two vectors are considered equal if they have the same magnitude and direction. They do not need to start at the same point. The magnitude of a vector A — that is, its length — is written as A, not in bold type.

The next figure shows the standard coordinate system for vectors. Note the x and y axes, which vectors are measured against.

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The x and y axes are measured using some standard physics units, such as centimeters. Positive x is to the right, negative x is to the left; positive y is up, negative y is down. The center of the graph, where the axes meet, is called the origin. A vector is commonly described by its length and its angle from the positive x axis (0 degrees to 360 degrees).

Sample question

  1. Suppose the vector is 3.0 centimeters long and at an angle of 45 degrees with respect to the x axis. How would you exactly describe this vector?

    The correct answer is 3.0 centimeters long and at an angle of +45 degrees with respect to the positive x axis.

Practice questions

  1. A marble starts at the origin and rolls 45 meters to the right. Describe where it ends up, in vector notation.

  2. A marble starts at the origin and rolls 45 meters to the right. Then it moves 90 meters to the left. Describe where it ends up, in vector notation.

Following are answers to the practice questions:

  1. 45 meters at 0 degrees

    The marble ends up 45 meters to the right of the origin, which is to say it ends up 45 meters from the origin, at 0 degrees.

  2. 45 meters at 180 degrees

    1. The marble moved 45 meters to the right at 0 degrees and then moved left for 90 meters.

    2. The marble ends up at 45 meters on the other side of the origin, at 45 meters at 180 degrees.