Question Video: Finding the Unknown Components of Three Forces given Their Resultant Mathematics

Video Transcript

Three forces, 𝐅 one equals negative five 𝐢 plus 10𝐣 newtons, 𝐅 two equals 𝑎𝐢 minus five 𝐣 newtons, and 𝐅 three equals negative four 𝐢 plus 𝑏𝐣 newtons act at a point. Their resultant is six root two newtons northwest. Determine the values of 𝑎 and 𝑏. Remember, the resultant is the vector sum of all the forces acting on an object.

In this question, we’re dealing with three forces: 𝐅 one, 𝐅 two, and 𝐅 three. Now, we’re going to need to be a little bit careful. Because not only have we been given their resultant, but we’ve been given the magnitude of the resultant and the direction in which this points. So we’re going to need to begin by finding the vector sum of our three forces; that’s 𝐅 one plus 𝐅 two plus 𝐅 three. That’s equal to negative five 𝐢 plus 10𝐣 plus 𝑎𝐢 minus five 𝐣 plus negative four 𝐢 plus 𝑏𝐣. It makes sense that we add together the components for 𝐢 and, separately, add together the components for 𝐣. That gives us negative five plus 𝑎 minus four 𝐢 plus 10 minus five plus 𝑏 𝐣. This simplifies to 𝑎 minus nine 𝐢 plus 𝑏 plus five 𝐣.

Now, we know the magnitude of this resultant to be six root two newtons. And we know this acts in a northwest direction. So let’s sketch this out. We can sketch this as a right-angled triangle as shown. In fact, since this is acting strictly in a northwest direction, we know that we have an isosceles triangle with two angles of 45 degrees. We know that the horizontal component for the force is 𝑎 minus nine and the vertical component is 𝑏 plus five. But since our triangle of forces, we said, is an isosceles triangle, we know that 𝑏 plus five and 𝑎 minus nine must be of equal magnitude. So we’re going to call them 𝑥 for now and use the Pythagorean theorem to work out the value of 𝑥.

The Pythagorean theorem tells us that the square of the hypotenuse is equal to the sum of the squares of the other two sides in the triangle. So 𝑥 squared plus 𝑥 squared must be equal to six root two squared. On the left, we get two 𝑥 squared. And on the right, we get 36 times two. Now, rather than evaluating 36 times two, let’s just divide through by two. And we see that 𝑥 squared is equal to 36. We then take the square root of both sides of our equation, remembering to take both the positive and negative square root of 36. So we get 𝑥 equals plus or minus six.

Now, we are going to need to take into account that 𝑥 could be both a positive and a negative value. And that’s because when we step away from the idea of this being a triangle, we realize that 𝑥 represents the component of the forces. And these can act in both a positive and negative direction. We’ll begin by considering the horizontal direction. Now, we know that that’s represented by 𝑎 minus nine. 𝑥 is six. But since this is acting in the negative 𝑥-direction, we would say that 𝑎 minus nine is equal to negative six. Then we simply solve this equation for 𝑎 by adding nine to both sides. And we find 𝑎 is equal to three.

We’ll repeat this process in the vertical direction. This time, we’re looking at the positive 𝑦-direction. So we’re going to say that 𝑏 plus five is equal to positive six. Then we solve for 𝑏 by subtracting five from both sides. And we find 𝑏 is equal to one. And so, we’ve determined the values of 𝑎 and 𝑏 based on the information in our question. 𝑎 is equal to three and 𝑏 is equal to one.