Pre-Calculus

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position (without rotating it), then the vector we obtain at the end of this process is the same vector we had in the beginning.

Two examples of vectors are those that represent force and velocity. Both force and velocity are in a particular direction. The magnitude of the vector would indicate the strength of the force or the speed associated with the velocity.

We denote vectors using boldface as in aa or bb. Especially when writing by hand where one cannot easily write in boldface, people will sometimes denote vectors using arrows as in a⃗a→ or b⃗b→, or they use other markings. We won't need to use arrows here. We denote the magnitude of the vector aa by ∥a∥∥a∥. When we want to refer to a number and stress that it is not a vector, we can call the number a scalar. We will denote scalars with italics, as in aa or bb.

You can explore the concept of the magnitude and direction of a vector using the below applet. Note that moving the vector around doesn't change the vector, as the position of the vector doesn't affect the magnitude or the direction. But if you stretch or turn the vector by moving just its head or its tail, the magnitude or direction will change. (This applet also shows the coordinates of the vector,