calculus-based physics for scientists and engineers

Vectors: Descriptive Notation

What is a vector?

For the purposes of this course, a vector is a quantity that requires a magnitude and a direction. The magnitude corresponds to the amount of the quantity and the direction is the ... um ... direction. A vector can be represented by an arrow. The length of the arrow can represent the magnitude, the amount of the quantity. The arrow points in the direction of the vector, because anything else would be silly.

Vectors in Descriptive Notation: Addition

We add vectors geometrically using the "tail-to-tip" method. (Which is too bad, because "tip-to-tail" flows off the tongue much better.) Translate (but not rotate) the second vector so the tail of the second vector is on the tip of the first vector. Then draw a line from the tail of the first to the tip of the second.

Examples of Adding Vectors Graphically

The best way to learn this is to see some examples. Brush up on your laws of Sines and Cosines!

Vectors in Descriptive Notation: Multiplication by a Scalar and Subtraction

A scalar is just a number. To multiply a vector by a scalar, note if the scalar is non-negative or negative. Minus signs are associated with directions with vectors.

  • If the scalar is non-zero, multiply the scalar and the magnitude of the original vector. The result is the magnitude of the resulting vector.
  • If the scalar is -1, then the resulting vector has the same magnitude but points in the opposite direction.
  • If the scalar is any other negative number do both of the above where the magnitude is positive.

To subtract vectors, take the second vector, multiply by -1 and add to the first. If you think about it, that is the same procedure you use to subtract numbers.


Do Now!   Do these exercises immediately.

Not Now!   Do these after you start to forget the topic, say in a week.

More!   More exercises if you want. Maybe review before a test.

Required Problems


1. A jogger runs east 3 km, then southeast 4.5 km. The jogger turns to an unknown direction northeast, until they arrive exactly 6.5 km east from where they started. What is the length, and in what direction east of north, was the final leg of the run?

Optional Problems



For the graph, \(+x\) is to the right, \(+y\) is up, and each tick is 1 unit.





  • \(\vec{a}\) has a magnitude of 6.32 and is pointing 18.4° right of \(+y\) axis.
  • \(\vec{b}\) has a magnitude of 4.12 and is pointing 14.0° below the \(+x\) axis.
  • \(\vec{c}\) has a magnitude of 3.00 and is pointing along the \(-y\) axis.
  • \(\vec{d}\) has a magnitude of 5.00 and is pointing 36.9° below the \(-x\) axis.

Using graphical methods, find the magnitude and direction of: \(\vec{h} = 3\vec{b}+2\vec{d}-4\vec{c}\).