### Video Transcript

Three forces, five π plus 10π
newtons, ππ 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?

Weβve been given the vector
representation of three forces in newtons acting on a single particle. Weβve also been given their
resultant. Thatβs the total of these three
forces. In order to work out the values of
π and π then, weβll 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. Thatβs five π plus 10π plus ππ
minus five π plus 15π plus π plus seven π. And to add vectors, we add their
individual components. So weβll add their horizontal
components. Thatβs five, π, and 15. And weβll add their vertical
components. Thatβs 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, weβll solve these
equations for π and π.

To solve this first equation, letβs
subtract 20 from both sides. We get π is equal to negative
two. And to solve the second equation,
weβll subtract 12 from both sides. And we get π is equal to
seven. And weβve 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.