Video: Finding the Unknown Components of Two Forces given the Components of the Resultant

The forces πβ = 2π’ + 2π£, πβ = ππ’ + 9π£, and πβ = 9π’ + ππ£ act on a particle, where π’ and π£ are two perpendicular unit vectors. Given the forcesβ resultant π = 2π’ β 6π£, determine the values of π and π.

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Video Transcript

The forces π one equals two π’ plus two π£, π two equals ππ’ plus nine π£, and π three equals nine π’ plus ππ£ act on a particle, where π’ and π£ are two perpendicular unit vectors. Given the forcesβ resultant π equal to two π’ minus six π£, determine the values of π and π.

Letβs begin by recalling what we actually mean by a resultant force. The resultant force is the overall force acting on the object. Here, itβs the sum of π one, π two, and π three. Now, weβre actually given that the resultant of these forces is two π’ minus six π£. Well, since the resultant is the sum of the three forces, we can say that π sub one plus π sub two plus π sub three must be equal to π. Weβre going to replace each force as given in the question. And we see that two π’ plus two π£ plus ππ’ plus nine π£ plus nine π’ plus ππ£ must be equal to two π’ minus six π£.

Next, weβre going to collect together the π’-components and, separately, the π£-components of each force on the left-hand side. The π’-components are two, π, and nine. And the π£-components are two, nine, and π. And so, we can say that two plus π plus nine π’ plus two plus nine plus π π£ must be equal to two π’ minus six π£. Now, of course, for the vector on the left-hand side to be equal to the vector on the right-hand side of our equation, we know that the individual components must themselves be equal. Equating the components in the π’-direction, and we get two plus π plus nine equals two.

Similarly in the π£-direction, our equation is two plus nine plus π equals negative six. We can simplify our first equation, and we get π plus 11 equals two. Then, we solve for π by subtracting 11 from both sides. And we find π to be equal to negative nine. Similarly, our other equation becomes 11 plus π equals negative six. To solve this equation for π, weβre going to subtract 11 from both sides. And we get π equals negative 17. So, given the information about our three forces and their resultant, we found π is equal to negative nine and π is equal to negative 17.

And in fact, we could check our answers by substituting π equals negative nine and π equals negative 17 back into our original forces. Doing that and then finding the sum of the three forces, we do indeed find that itβs equal to two π’ minus six π£, which we saw is equal to π.