Engineering Topics – Vectors

As an ex-engineer, I know that almost all of the math skills I have learned in school have been used in my career. However, students generally do not appreciate this when, for example, they are factoring a quadratic. It becomes easier to demonstrate the usefulness of math to students when they are being taught skills that have a direct application. So I thought I would present a few topics in maths with some applications in engineering. The first of these topics is vectors.

A vector is an object with a magnitude and a direction. If you are travelling at 60 km/hr, that number is just a magnitude. But if you are travelling north at 60 km/hr, that has a directions as well and can be represented as a vector.

There are a lot of new math skills associated with vectors, but I will only cover what is necessary to present this application of vectors.

Vector Addition

Vectors can be added algebraically but I will just be talking about how to add vectors graphically.

Here is a typical vector:

It’s magnitude can be its physical length or a labelled quantity like 60 km/hr or 100 N (newtons) of force. Its direction though, needs a reference. This will usually be a coordinate system like the Cartesian coordinate system:

This vector represents a velocity of 60 km/hr in the direction 30° from the positive x-axis. Now this direction is completely dependent on the coordinate system used. We could have made the x-axis line up with the vector itself. But once established, questions regarding the vector will be answered with respect to the coordinate system used.

A vector can be the sum of two or more vectors. There are many variations to this but the scenario applicable to this post is where the vector is the sum of two vectors that are perpendicular to each other. Given the coordinate system we have used, the vector V can be broken up into two components, Vx and Vy. That is, V = Vx + Vy. Graphically, this looks like:

Now we already know the direction with respect to this coordinate system of each of the component vectors Vx and Vy, but what are their magnitudes?

Looking at the above figure, you can see that the vector V splits the rectangle formed by the x and y axes and the dotted lines into two similar right triangles. If the length of the hypotenuse of these triangles is 60, then

\[\left|{\boldsymbol{V_x}}\right|=60\text{ cos}(30°)=51.96 \text{ km/hr}\]

where |vector| is notation to represent the magnitude of a vector. Similarly, you can see that

\[ \left|{\boldsymbol{V_y}}\right|=60\text{ sin}(30°)=30 \text{ km/hr}\]

If you were given the perpendicular vector components, you can use the Pythagorean theorem to find the magnitude. Let’s use the components found above to see if they give back the original vector:

\[\left|{\boldsymbol{V}}\right|=\sqrt{\boldsymbol{V_x}^2+\boldsymbol{V_y}^2}=\sqrt{51.96^2+30^2}\approx60\text{ km/hr}\]

This is approximately 60 because the 51.96 number used is a two decimal place approximation to the actual irrational number.

So here is a practical application of this:

A cable supports a beam at an angle of 29.745°. A weight at the end of the beam provides a force of 500 N. Assume that the weight of the beam is negligible. What is the tension (force) seen by the cable?

Now before I tackle this, a bit of background. First, the “N” in “500 N” stands for “newtons”. This is the unit of force in the SI system of units. SI is the International System of Units that is usually used in science and engineering. Though your mass may be 80 kg, the force you exert on the earth is 784.8 N. In return, the earth exerts a force of 784.8 N against you in the opposite direction. Second, this problem is a statics problem, that is, nothing is moving. That means that all involved forces (vector quantities that have direction) must add to zero. The force you exert on the earth and the opposite force that the earth exerts on you add to zero, so there is no motion up or down. Third, tension and compression in a structural member is a force that the member exerts on neighbouring objects in response to forces that cause the tension or compression.

Let’s isolate the end of this structure where the force is being applied:

As there is no motion, there must be an opposite but equal force of 500 N:

What is causing this opposite force? It’s the tension in the cable. This tension must be great enough to cause a force of 500 N in the y (up) direction. This 500 N up is the y component of the force from the cable:

Knowing the y component, the opposite side of the right triangle shown, we can use trigonometry to find the hypotenuse, the value of the tension in the cable T:

\[\left|{\boldsymbol{T}}\right|=\frac{500}{\text{sin}(29.745°)}=1000.78 \text{ N}\]

I leave it to you to review the definition of the sine of an angle in a right triangle.

Even though this was not asked, the tension in the cable creates a force in the x direction (horizontal direction). Again using trigonometry, the value of this force is 875.00 N. Where is the force that opposes this so that there is no motion? It comes from the compression imposed on the horizontal member:

This is an engineering problem because if you were to design and build this structure, you would have to ensure that the materials used could handle the tension and compression loads with an additional safety factor. Before reading this post, if you just thought that the cable needs to handle 500 N, you would soon be looking for a new job.