Three forces, five 𝑖 plus 10𝑗
newtons, 𝑎𝑖 minus five 𝑗 newtons, and 15𝑖 plus 𝑏 plus seven 𝑗 newtons, act on
a particle. Given that the resultant of the
forces is 18𝑖 plus 19𝑗 newtons, what are the values of 𝑎 and 𝑏?
We’ve been given the vector
representation of three forces in newtons acting on a single particle. We’ve also been given their
resultant. That’s the total of these three
forces. In order to work out the values of
𝑎 and 𝑏 then, we’ll need to use this information to set up equations in terms of
𝑎 and 𝑏.
And since the resultant is the sum
total of the three forces, we begin by adding them. That’s five 𝑖 plus 10𝑗 plus 𝑎𝑖
minus five 𝑗 plus 15𝑖 plus 𝑏 plus seven 𝑗. And to add vectors, we add their
individual components. So we’ll add their horizontal
components. That’s five, 𝑎, and 15. And we’ll add their vertical
components. That’s 10, negative five, and 𝑏
Five plus 𝑎 plus 15 simplifies to
20 plus 𝑎. And 10 minus five plus 𝑏 plus
seven simplifies to 12 plus 𝑏. So we have the horizontal component
to be 20 plus 𝑎 and the vertical component to be 12 plus 𝑏.
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 𝑎
must be equal to 18. And 12 plus 𝑏 must therefore be
equal to 19. And then, we’ll solve these
equations for 𝑎 and 𝑏.
To solve this first equation, let’s
subtract 20 from both sides. We get 𝑎 is equal to negative
two. And to solve the second equation,
we’ll subtract 12 from both sides. And we get 𝑏 is equal to
seven. And we’ve set out what we needed to
do. We found the values of 𝑎 and
𝑎 is equal to negative two. And 𝑏 is equal to seven.