### Video Transcript

In the figure, ๐จ๐ฉ is a rod fixed to a vertical wall at end ๐ด. The other end ๐ต is connected to a wire ๐ฉ๐, where ๐ถ is fixed to a different point on the same vertical wall. If the tension in the wire equals 65 newtons, calculate the moment of the tension about point ๐ด in newton meters.

Weโve been provided with a diagram of the system on a set of 3D axes. The ๐ฅ-axis is pointing out of the screen, the ๐ฆ-axis is pointing to the right, and the ๐ง-axis is pointing up. Here, we can see the rod with one end attached to the wall as described. And we can see that this wire connects the other end of the rod to the wall as well. So, the diagram only shows us what appears to be two pieces of wall floating in space. But the question makes it clear that both the wire and the rod are connected to the same vertical wall. So, the two walls shown in the diagram are actually parts of the same wall.

There are three specific points that are labeled in the diagram and also mentioned in the question: ๐ด, ๐ต, and ๐ถ. ๐ด is the point where one end of the rod is connected to the wall, and this is also the origin about 3D axes. ๐ต is where the other end of the rod is connected to the wire and ๐ถ is where the other end of the wire is connected to the wall.

The question asks us to calculate the moment of the tension about point ๐ด in newton meters. In other words, we want to calculate the moment about point ๐ด, which is produced by the tension in the wire. And to do this, we need to use this equation. This equation tells us that the moment vector ๐ is equal to the cross product of a displacement vector ๐ and a force vector ๐
. Specifically, ๐
is just the force vector that produces the moment. So, in this question, thatโs the force vector acting at ๐ต due to the tension in the wire. And that ๐ is the displacement vector of the point at which the force acts relative to the point that we want to calculate the moment about.

So in this question, because the force is acting at point ๐ต and we want to calculate the moment about point ๐ด, the vector ๐ is the displacement vector that would take us from ๐ด to ๐ต. In other words, itโs the vector ๐จ๐ฉ. In this question, we havenโt been given the components of ๐ or ๐
. So in order to work out their cross product, we need to find out their components from the information given in the question. Letโs start by finding the components of the vector ๐.

Now, we previously noticed that ๐ is equal to the vector ๐จ๐ฉ, where ๐ด is located at the origin and ๐ต is located 12 meters away in the positive ๐ฆ-direction. So, the vector ๐จ๐ฉ has a magnitude of 12 meters and points in the positive ๐ฆ-direction. Equivalently, we could say that the components of this vector in the ๐ฅ- and ๐ง-direction is zero and the component in the ๐ฆ-direction is 12. Written as a vector in meters, this gives us zero ๐ข plus 12๐ฃ plus zero ๐ค, and in its simplest form, thatโs just 12๐ฃ.

Okay, so next, we just need to find the components of the force vector ๐
. Then we can calculate the cross product of ๐ and ๐
, and this will give us the moment vector ๐. Unfortunately, finding the vector ๐
is a bit trickier than finding ๐. Now, ๐
describes the force vector which is acting at ๐ต due to the tension in the wire. This means that we know two things about the vector ๐
.

Firstly, the magnitude of ๐
is 65 newtons, as weโre told that thereโs 65 newtons of tension in the wire. And secondly, because the wire goes from ๐ต to ๐ถ, we know that the force acting at ๐ต must go along the line ๐ฉ๐. In other words, the vector ๐
is parallel to ๐ฉ๐. Itโs useful to remind ourselves at this point that even though ๐
acts along ๐ฉ๐, itโs not equal to ๐ฉ๐. ๐
is a force vector, and ๐ฉ๐ is a displacement vector. So, the magnitude of ๐
has nothing to do with the magnitude of ๐ฉ๐.

We can make this a bit clearer in our diagram by drawing the vector ๐
in a different color to the wire. Now, these two facts actually describe ๐
completely. After all, we have its magnitude and its direction. However, we need to calculate the actual components of ๐
in the ๐ฅ-, ๐ฆ-, and ๐ง-direction so that we can calculate ๐. In order to find these components, we first need to better describe the direction that ๐
points in. We said that itโs parallel to ๐ฉ๐, but what are the components of ๐ฉ๐? We can figure these out by looking at the measurements in the diagram.

To get from point ๐ต to point ๐ถ, weโd need to travel 12 meters in the negative ๐ฆ-direction, three meters in the positive ๐ง-direction, and then four meters in the negative ๐ฅ-direction. So expressed in meters, the vector ๐ฉ๐ has an ๐ฅ-component of negative four ๐ข, a ๐ฆ-component of negative 12๐ฃ, and a ๐ง-component of three ๐ค. So, ๐
is parallel to this vector, but we know it has a magnitude of 65 newtons.

At this point, itโs useful to remind ourselves that if ๐
is parallel to ๐ฉ๐, that means we can obtain ๐
by scaling ๐ฉ๐. In other words, since the two vectors point in the same direction, then if we stretch the vector ๐ฉ๐ or scale it by the right amount, then it will be equal to ๐
. Mathematically, we do this by multiplying the vector ๐ฉ๐ by some scalar constant ๐ค, in other words a number. So, we can obtain ๐
by multiplying the vector ๐ฉ๐ by a number so that its magnitude becomes 65. But what number is this? Whatโs the scalar constant that we need to multiply ๐ฉ๐ by to give us ๐
?

One way we can solve this problem is to find the unit vector that points in the direction of ๐ฉ๐. In this case, letโs call this unit vector ๐ฎ denoted with a hat symbol to show that itโs a unit vector. Remember that a unit vector has a magnitude of one. If we can find this unit vector, then multiplying it by 65 will tell us the vector that points in the same direction and has a magnitude of 65. In other words, it will tell us ๐
. Now, fortunately, finding the unit vector that points in the direction of some vector is relatively straightforward. Itโs done by simply dividing that vector by its own magnitude. That means ๐ฎ, the unit vector that points in the direction of ๐ฉ๐, is given by dividing the vector ๐ฉ๐ by the magnitude of ๐ฉ๐.

We can calculate the magnitude of ๐ฉ๐ by using the three-dimensional form of Pythagorasโs theorem. Itโs given by the square root of the ๐ฅ-component squared plus the ๐ฆ-component squared plus the ๐ง-component squared. Simplifying the denominator of this expression, negative four squared is 16, negative 12 squared is 144, and three squared is nine. 16 plus 144 plus nine is 169, and the square root of 169 is 13.

So, weโve shown that because ๐ฉ๐ has a magnitude of 13, dividing it by 13 gives us the unit vector that points in the direction of ๐ฉ๐. Since weโve shown that ๐
is equal to 65 times this unit vector ๐ฎ, this means that ๐
is equal to 65 times ๐ฉ๐ over 13, which we can equivalently write as 65 over 13 times ๐ฉ๐. In other words, scaling ๐ฉ๐ by a factor of 65 over 13 gives us ๐
.

Note that weโre also technically changing the units from meters to newtons. When we multiply the vector ๐ฉ๐ by a factor of 65 over 13, we effectively multiply each of the components of ๐ฉ๐ by 65 over 13. This gives us an ๐ฅ-component of negative four ๐ข times 65 over 13, a ๐ฆ-component of negative 12๐ฃ times 65 over 13, and a ๐ง-component of three ๐ค times 65 over 13. Note that 65 over 13 simplifies to just five. So, weโre really multiplying each of these components by five.

We can now simplify this one term at a time. Negative four times 65 over 13 or negative four times five is negative 20, leaving us with an ๐ฅ-component of negative 20๐ข. Looking at the next term, we have negative 12๐ฃ times 65 over 13. This is equal to negative 12๐ฃ times five, which is negative 60๐ฃ. And finally, three ๐ค times 65 over 13 is three ๐ค times five, which is equal to 15๐ค. So, there we have it. We knew that ๐
had a magnitude of 65 newtons and pointed in the same direction as ๐ฉ๐. So, we found ๐
by calculating the unit vector in the direction of ๐ฉ๐ and then multiplying this by 65.

Now that we found the components of the displacement vector ๐ and the force vector ๐
, we can calculate the moment vector thatโs produced by ๐
by working out the cross product of ๐ and ๐
. To do this, we calculate this three-by-three determinant where the elements in the top row are the unit vectors ๐ข, ๐ฃ, and ๐ค. The elements in the middle row are the ๐ฅ-, ๐ฆ-, and ๐ง-components of the displacement vector ๐, written without their unit vectors. And the elements in the bottom row are the ๐ฅ-, ๐ฆ-, and ๐ง-components of the force vector ๐
, also written without their unit vectors.

๐ has an ๐ฅ-component of zero, a ๐ฆ-component of 12, and a ๐ง-component of zero. And ๐
has an ๐ฅ-component of negative 20, a ๐ฆ-component of negative 60, and a ๐ง-component of 15. This determinant is then calculated effectively in three parts. First, we have the unit vector ๐ข multiplied by 12 times 15 minus zero times negative 60. Next, we subtract the unit vector ๐ฃ multiplied by zero times 15 minus zero times negative 20. And finally, we add the unit vector ๐ค multiplied by zero times negative 60 minus 12 times negative 20.

Okay, now, we just need to simplify each term. Looking at the ๐ข-term, we have 12 times 15 which is 180. And weโre subtracting zero times negative 60, which is of course zero. So the ๐ข-term simplifies to 180๐ข. Looking at the ๐ฃ-term, we have zero times 15, which is zero. And weโre subtracting zero times negative 20, which is also zero. This means that this term simplifies to negative ๐ฃ times zero, which is of course just zero. So, we donโt need to write this term down. Finally, looking at the ๐ค-term over here, we have zero times negative 60 which is zero. And then, weโre subtracting 12 times negative 20. 12 times negative 20 is negative 240. So, we have zero minus negative 240, which is just 240. So, this term simplifies to 240๐ค.

So, because weโve now found the cross product of ๐ and ๐
, that means that this is equal to the moment vector ๐. We can also note that because we expressed the displacement vector ๐ in meters and the force vector ๐
in newtons, then weโve calculated a moment vector in newton meters as specified in the question. So, this is our final answer. The moment about point ๐ด which is produced by the tension shown in the diagram expressed in newton meters is 180๐ข plus 240๐ค.