### Video Transcript

Given that the vector π
sub one is equal to eight π’ minus five π£ and the vector π
sub two is equal to negative 15π’ minus five π£, and their resultant the vector π is negative ππ’ minus ππ£, determine the values of π and π.

In this question, weβre given two vectors π
sub one and π
sub two. Weβre given these in terms of the unit directional vectors π’ and π£. And itβs worth pointing out here because our vectors are labeled π
sub one and π
sub two, we can assume that these represent forces. Weβre told that the resultant of these two forces is given in the form negative ππ’ minus ππ£. We need to determine the values of π and π. To answer this question, letβs 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, itβs 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 π
one and the force π
two. And to combine two forces as vectors, we need to add them together. In other words, the resultant vector π is going to be equal to π
sub one plus π
sub two. So we start with the equation π is equal to π
sub one plus π
sub two. But weβre given these vectors in terms of the unit directional vectors, π’ and π£. We know that π is equal to negative ππ’ minus ππ£. We know that π
sub one is equal to eight π’ minus five π£. And we know that π
sub two is negative 15π’ minus five π£. So we can just substitute these in, which gives us the following equation.

We want to use this equation to find the values of π and π. To simplify the right-hand side of this equation, we need to add together our π’-terms and our π£-terms. First, eight minus 15 is equal to negative seven. So our π’-term is going to be negative seven π’. Next, we have negative five π£ minus five π£. Well, negative five minus five is equal to negative 10. So our π£-term is negative 10π£. And we know this needs to be equal to our resultant vector, negative ππ’ minus ππ£. And for two vectors to be equal, recall we need that their components are the same. So negative π needs to be equal to negative seven and negative π needs to be equal to negative 10. This gives us two equations, and we can use these to find the values of π and π.

To find the value of π, we multiply the top equation through by negative one. Doing this, we see that π is equal to seven. We can do something very similar to find the value of π; we multiply our bottom equation through by negative one. And we see that our value of π is equal to 10. And this gives us our final answer π is equal to seven and π is equal to 10. Therefore, we were able to show if we have two forces represented by vectors β the first force, eight π’ minus five π£, and the second force, negative 15π’ minus five π£ β and the resultant vector is of the form negative ππ’ minus ππ£, then the value of π must be seven and the value of π must be 10.