Units and Dimensions

As we go through all of the physics material we will be talking about dimensions as well as units. It isn’t strictly necessary to memorize a ton of information about units and dimensions. However, having a working understanding of what they are is helpful as we work through the physics material on the MCAT as well as calculation questions.

Dimension Versus Units

So what is a dimension then and how is it different than a unit? A dimension is a descriptive term such as length, height, mass, or force without a numerical value. So we might ask how long is something or how much force did it exert? Here we are talking in terms of dimensions.

However, when we answer that question and quantify a specific value for length such as 10 m we move into the world of units. Which allows us to describe the specific value by using meters for length and height, kilograms for mass, and Newtons for force among others

While understanding this difference might not seem terribly useful. I like to have an awareness of both the dimensions and units when approaching calculation questions. Since we can use both to pick the correct equation or construct an equation if we don’t know one. Which we will look at in a later topic.

Hector as a snake has a very clear direction that they point in. Remember that Hector is the vector.

Vector or Scalar

Before we do that we should also understand that dimensions can fall into one of two categories vectors or scalars. If a dimension is a vector then it will have a numerical value and a direction associated with it. Take velocity for example, when we say the velocity of something is 30m/s we are saying that it is traveling a certain distance per unit time and must also posit a direction for that travel such as Southeast.

By comparison, speed is a scalar and has no direction attached to it. So I might say I was traveling at a speed of 30m/s. This tells you only how fast I was traveling but not the direction.

Taylor is a sleeping squirrel, which way they point is difficult to determine. Towards sleep town I suppose. Remember that Taylor is the scalar.

Vector Addition

Unlike scalars, vectors can be added to one another. Imagine Hector is swimming directly across a lake at 3m/s. If the lake is completely still then they will end up across the bank at the same horizontal point as they started.

However, if the lake is replaced with a river flowing downwards at 4m/s then they will be taken along for a ride as they swim and end up lower than where they started. In this case two vectors, velocity specifically, have been added together.

If we wanted to predict and actually calculate where Hector ended up we could using vector addition. The key here is aligning our vectors appropriately by placing them “tip” to “tail”.

Once these vectors are connected we can draw in the resultant vector aligning our vector tail to tail and tip to tip. From here it is a bit of math and potentially trigonometry, although the MCAT tends to shy away from sin, cos, and tan because we don’t have access to a calculator. Instead, we will almost always be dealing with a right triangle (90°) and can use The Pythagorean Theorem to solve for the resultant-vector-hypotenuse.

This happens to be a special 3:4:5 triangle so I don’t need to use The Pythagorean Theorem, but if this wasn’t a special triangle I would need to.

Vector Angle

Vectors can also be expressed in terms of their angle of intersection. In the above example the vectors were 90° apart from one another and hence formed a right triangle.

There are a couple of other angles of note on the MCAT, 0°/360°, 90°/270°, and 180°. For vectors that intersect at either 0° or 360°, they will be maximally additive.

While 270° is another way of denoting a right triangle situation.

Lastly vectors at 180° from one another are completely opposed and will cancel out maximally.

In this case the vectors are the same size and completely opposed resulting in no resultant vector.