Question Video: Analysis of Three Coplanar Forces in the Vector Form Acting on a Particle Together to Produce a Resultant | Nagwa Question Video: Analysis of Three Coplanar Forces in the Vector Form Acting on a Particle Together to Produce a Resultant | Nagwa

# Question Video: Analysis of Three Coplanar Forces in the Vector Form Acting on a Particle Together to Produce a Resultant Mathematics • Second Year of Secondary School

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Three forces, (5π + 10π) N, (ππ β 5π) N, and (15π + (π +7)π) N, act on a particle. Given that the resultant of the forces is (18π + 19π) N, what are the values of π and π?

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### Video Transcript

Three forces, five π plus 10π newtons, ππ minus five π newtons, and 15π plus π plus seven π newtons, act on a particle. Given that the resultant of the forces is 18π plus 19π newtons, what are the values of π and π?

Weβve been given the vector representation of three forces in newtons acting on a single particle. Weβve also been given their resultant. Thatβs the total of these three forces. In order to work out the values of π and π then, weβll need to use this information to set up equations in terms of π and π.

And since the resultant is the sum total of the three forces, we begin by adding them. Thatβs five π plus 10π plus ππ minus five π plus 15π plus π plus seven π. And to add vectors, we add their individual components. So weβll add their horizontal components. Thatβs five, π, and 15. And weβll add their vertical components. Thatβs 10, negative five, and π plus seven.

Five plus π plus 15 simplifies to 20 plus π. And 10 minus five plus π plus seven simplifies to 12 plus π. So we have the horizontal component to be 20 plus π and the vertical component to be 12 plus π.

Now we know that the actual horizontal component of the resultant of the forces is 18. And the actual vertical component is 19. We can say then that 20 plus π must be equal to 18. And 12 plus π must therefore be equal to 19. And then, weβll solve these equations for π and π.

To solve this first equation, letβs subtract 20 from both sides. We get π is equal to negative two. And to solve the second equation, weβll subtract 12 from both sides. And we get π is equal to seven. And weβve set out what we needed to do. We found the values of π and π.

π is equal to negative two. And π is equal to seven.

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