### Video Transcript

The diagram shows a system of forces measured in newtons. Given that π΄π΅ is equal to 24 centimeters and π΄π· is equal to 18 centimeters, determine π
, the magnitude of their resultant, and find π, the angle between their resultant and the positive π₯-axis. Give to the nearest minute.

Weβre dealing with three forces here. Letβs call these π
one, π
two, and π
three. We know that weβre working with two directions, so weβre going to label these as vector forces. Weβre also going to define the horizontal unit vector π’ and the vertical unit vector π£. This means we can call π
one negative 40π’. Now, weβve defined it as negative 40 because itβs acting in the negative direction. Similarly, π
two is acting downwards, so itβs negative 30π£. But what do we do about force three? We know its magnitude is 24 newtons. Weβre going to need to split it into both its horizontal and vertical components.

Now, to do so, weβll use the fact that π΄π΅ is equal to 24 centimeters and π΄π· is 18 centimeters. Now, of course, if π΄π· is 18 centimeters, π΅πΆ must also be 18 centimeters, so we draw a little right-angle triangle as shown. Weβre essentially looking to find the ratio between the hypotenuse and the other two sides. And so, letβs call this angle π. We have the opposite side of this triangle as 18 centimeters and the adjacent as 24. And so, we use the tan ratio. We get tan π equals opposite over adjacent, 18 over 24, which simplifies to three-quarters.

We know the triangle stays in proportion with force three, so we relabel our triangle with the hypotenuse as being 24 newtons. This time, letβs label the adjacent π₯ and the opposite π¦. We can use the cos ratio to find expressions linking π and π₯. When we do, we get cos π is adjacent over hypotenuse, π₯ over 24. And so, π₯ is 24 cos π. Similarly, we also find that sin π is π¦ over 24. And we can rearrange and write π¦ equals 24 sin π.

Now, we go back to the fact that tan π is equal to three-quarters. In this case, we actually have a Pythagorean triple. We know that if the opposite is three centimeters or three units, the adjacent is four units, then the hypotenuse is five units. Then, cos π is four-fifths; itβs adjacent over hypotenuse. And sin π is three-fifths; itβs opposite over hypotenuse. So, we find π₯ is equal to 24 times four-fifths. And π¦ is equal to 24 times three-fifths. This gives us ninety-six fifths and seventy-two fifths as the horizontal and vertical components, respectively. And since both the horizontal and vertical components are acting in the positive direction, force three is 96 over five π’ plus 72 over five π£.

Weβre now going to clear some space for the next step. Weβre told to find π
, the magnitude of the resultant of these forces. Now, the resultant is the vector sum of all forces. So, we can say that the resultant vector is π
one plus π
two plus π
three. To add the vectors, we add their horizontal and vertical components. Well, the horizontal components are negative 40 and 96 over five. And the vertical components are negative 30 and 72 over five. This simplifies to negative one hundred and four fifths π’ minus seventy-eight fifths π£. And so, the resultant force acts in this direction shown. In the horizontal direction, itβs one hundred and four fifths newtons. And in the vertical direction, itβs seventy-eight fifths newtons.

The magnitude of the resultant is basically the length of the vector, so we use the Pythagorean theorem. Itβs the square root of seventy-eight fifths squared plus one hundred and four fifths squared. Thatβs 26 or 26 newtons. Now, weβre also looking to find π, the angle between the resultant and the positive π₯-axis. So, weβll begin by finding the angle πΌ. Thatβs the angle that π
makes with the negative π₯-axis. And so, weβre going to use the tan ratio. Tan πΌ is seventy-eight fifths over one hundred and four fifths. That simplifies to 78 over 104. And therefore, πΌ is the inverse tan of 78 over 104, which is 36.86 and so on degrees.

Now, of course, weβre measuring this from the positive π₯-axis, and so we as usual move in a counterclockwise direction. We therefore need to add 180 degrees to our value for πΌ. π is 180 plus 36.86, which is 216.86 and so on. Correct to the nearest minute, thatβs 216 degrees and 52 minutes.

And so, we found the magnitude of the resultant of these forces π
to be 26 newtons and the angle π it makes with the positive π₯-axis to be 216 degrees and 52 minutes.