Vectors

3 Vectors

In the previous section we learned to use a coordinate system to describe the location of an object. However, vectors provide an even better way of doing this.

A displacement vector is a set of instructions of how to move.

For example, the vector $\vec A$ could be “move $A_x$ m to the right, and $A_y$ m upwards,” which we can write as $\vec A=(A_x,A_y)$ . If $A_x$ is negative, then we move left instead of right. If $A_y$ is negative, we move down instead of up. The quantities $A_x$ and $A_y$ are called the components of the vector $\vec A$ . At this point, it sounds like we’re reinventing the wheel, but we’ll see in the next chapter why thinking about positions as vectors (rather than just coordinates) is so useful. For now, let’s consider some properties of vectors.

Exercise 3.1: Vector Representations

We can represent a displacement vector as an arrow that tells us how to move. E.g. the plot below shows three displacement vectors:

$\vec A$ = “move 3 to the right and 1 down.”
$\vec B$ = “move 1 to the left and 2 down.”
$\vec C$ = “move 3 to the right and 2 up.”

Drag each of the labels for vectors $\vec A$ , $\vec B$ , and $\vec C$ to the correct arrow.

The fact that we can represent vectors with arrows makes it obvious that the way in which a vector can give instructions is not unique! E.g. rather than specifying how far we move along the $x$ and $y$ axis, we can specify the direction and total distance travelled (e.g. travel 5 mi North-East). In math, the length of a vector (how far we need to go) is called the magnitude.

All vector quantities have both a magnitude and a direction.

The magnitude of a vector $\vec A$ is denoted as $|\vec A|$ . Since I find writing $|\vec A|$ annoying, I will use the shorthand notation $A$ for the magnitude of $\vec A$ . This is actually fairly common, so it’s good to know about it anyways.

Exercise 3.2: Calculating Vector Magnitudes

The plot below show the displacement vectors that would take you to see each of your three friends, Aisha ( $\vec A$ ), Beth ( $\vec B$ ), and Charlie ( $\vec C$ ). Each grid cell is 1 m across.

Use the diagram below to calculate the magnitude of the vectors $\vec A$ , $\vec B$ , and $\vec C$ . If you are feeling stuck, look at the hint.

We can use the same “trick” of looking at the right triangle formed by the vector and the $x$ and $y$ axis to calculate the components of a vector given its magnitude. Let’s try it!

Exercise 3.3: Calculating Vector Components

Consider the vector $\vec r = (r_x,r_y)$ shown in the picture below, and let $r$ be the vector’s magnitude.

A) Find the $x$ and $y$ components of the vector $\vec r$ in terms of the magnitude $r$ and the angle $\theta$ .

B) Repeat, but use the angle $\phi$ instead of $\theta$ . There’s a very good reason I’m asking you to do this, which we’ll get to in a second.

If you’re not sure how to do part B, we go through the solution in the video below.

WARNING: If you have taken physics before, you might have memorized something like $r_x=r\cos \theta$ . Do not do this! As you saw in the previous problem, which trigonometric function you use to find $r_x$ (or $r_y$ ) depends on what angle you are using! It is best to just look at the drawing, and figure it out.

Key Takeaways

3 Vectors

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