Given that the vector 𝐅 sub one is equal to eight 𝐢 minus five 𝐣 and the vector 𝐅 sub two is equal to negative 15𝐢 minus five 𝐣, and their resultant the vector 𝐑 is negative 𝑎𝐢 minus 𝑏𝐣, determine the values of 𝑎 and 𝑏.
In this question, we’re given two vectors 𝐅 sub one and 𝐅 sub two. We’re given these in terms of the unit directional vectors 𝐢 and 𝐣. And it’s worth pointing out here because our vectors are labeled 𝐅 sub one and 𝐅 sub two, we can assume that these represent forces. We’re told that the resultant of these two forces is given in the form negative 𝑎𝐢 minus 𝑏𝐣. We need to determine the values of 𝑎 and 𝑏. To answer this question, let’s start by recalling exactly what we mean by the resultant of two or more forces. Sometimes we only have one force acting on an object. However, it’s possible to have two or more forces acting on the same object.
The resultant force will then be the force we find by considering all of the forces acting on our body. What this really means is, in this case, we have two forces acting on our object, the force 𝐅 one and the force 𝐅 two. And to combine two forces as vectors, we need to add them together. In other words, the resultant vector 𝐑 is going to be equal to 𝐅 sub one plus 𝐅 sub two. So we start with the equation 𝐑 is equal to 𝐅 sub one plus 𝐅 sub two. But we’re given these vectors in terms of the unit directional vectors, 𝐢 and 𝐣. We know that 𝐑 is equal to negative 𝑎𝐢 minus 𝑏𝐣. We know that 𝐅 sub one is equal to eight 𝐢 minus five 𝐣. And we know that 𝐅 sub two is negative 15𝐢 minus five 𝐣. So we can just substitute these in, which gives us the following equation.
We want to use this equation to find the values of 𝑎 and 𝑏. To simplify the right-hand side of this equation, we need to add together our 𝐢-terms and our 𝐣-terms. First, eight minus 15 is equal to negative seven. So our 𝐢-term is going to be negative seven 𝐢. Next, we have negative five 𝐣 minus five 𝐣. Well, negative five minus five is equal to negative 10. So our 𝐣-term is negative 10𝐣. And we know this needs to be equal to our resultant vector, negative 𝑎𝐢 minus 𝑏𝐣. And for two vectors to be equal, recall we need that their components are the same. So negative 𝑎 needs to be equal to negative seven and negative 𝑏 needs to be equal to negative 10. This gives us two equations, and we can use these to find the values of 𝑎 and 𝑏.
To find the value of 𝑎, we multiply the top equation through by negative one. Doing this, we see that 𝑎 is equal to seven. We can do something very similar to find the value of 𝑏; we multiply our bottom equation through by negative one. And we see that our value of 𝑏 is equal to 10. And this gives us our final answer 𝑎 is equal to seven and 𝑏 is equal to 10. Therefore, we were able to show if we have two forces represented by vectors — the first force, eight 𝐢 minus five 𝐣, and the second force, negative 15𝐢 minus five 𝐣 — and the resultant vector is of the form negative 𝑎𝐢 minus 𝑏𝐣, then the value of 𝑎 must be seven and the value of 𝑏 must be 10.