### Video Transcript

Two parallel forces have magnitudes of 24 newtons and 60 newtons as shown in the figure. The distance between their lines of action is 90 centimetres. Given that the two forces are acting in opposite directions, determine the resultant π
and the distance π₯ between its line of action and point π΄.

Weβre told in the problem statement and shown in the diagram the magnitude of the two forces involved. Weβll call the first force πΉ sub π΄ and the second πΉ sub π΅. Knowing that the distance between π΄ and π΅ is 90 centimetres, we want to determine two things. We want to calculate π
, the resultant of the forces πΉ sub π΄ and πΉ sub π΅, and we also want to calculate π₯, where π₯ is the distance from point π΄ to the line of action of the resultant force π
.

Letβs start out by calculating the resultant π
. As we look at our diagram, we see that positive force direction has been defined as up. So we can write that π
equals πΉ sub π΄ minus πΉ sub π΅. Entering in the values of those two forces, we find that π
is equal to negative 36 newtons. Thatβs the resultant force of the two original forces.

Now, we want to move on to calculating π₯. Given the relative magnitudes of πΉ sub π΄ and πΉ sub π΅, we can expect the point π₯ to exist out on the line to the right of the segment π΄π΅, where π₯ is measured from point π΄. If π₯ identifies the line of action of our resultant force π
, then we can write πΉ sub π΄ times π₯ minus πΉ sub π΅ times π₯ minus 90 centimetres is equal to zero.

We can group the terms involving π₯ on the left-hand side of our equation, subtract πΉ sub π΅ times 90 centimetres from both sides, and then divide both sides by πΉ sub π΄ minus πΉ sub π΅, cancelling that term on the left. We know the values for πΉ sub π΄ and πΉ sub π΅ and can plug those in now. When we calculate this fraction, we find that π₯ is equal to 150 centimetres. Thatβs the distance between point π΄ and the line of action of the resultant force π
.