Vectors

Vectors are a powerful mathematical tool for representing many physical entities. Understanding their meaning and becoming comfortable using them will make many physics concepts much easier to grasp!

Just like numbers, vectors can be represented using variables. To distinguish variables that represent numbers and variables that represent vectors, we introduct a little arrow. Here we are saying that rr is a variable representing the number 5:

r=5r = 5

And here we are saying that rr is a vector:

r⃗=5ı^+5ȷ^\vec{r} = 5 \hat{\imath} + 5 \hat{\jmath}

Don’t worry about this weird ı^\hat{\imath} and ȷ^\hat{\jmath}! We’ll get to them!

Vectors can also help simplify our equations! Here is Newton’s Second Law in 3 dimensions without using vectors:

∑Fx=max∑Fy=may∑Fz=maz\begin{align*} \sum F_x &= m a_x \\ \sum F_y &= m a_y \\ \sum F_z &= m a_z \end{align*}

Using vectors, we can turn these three equations into one:

∑F⃗=ma⃗\sum \vec{F} = m \vec{a}

Definition of a Vector

For us phycists, vectors can be though of as line segments that have a direction and a length. They are most commonly represented geometrically as an arrow, but it’s important to realize that we only care about the direction and length of the arrow, not its position.

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1.1 Vectors and Kinematics Next →
2.1 Newton's Laws