### Video Transcript

Two parallel forces, πΉ one and πΉ two, are acting at two points, π΄ and π΅, respectively, in a perpendicular direction on line segment π΄π΅. π΄π΅ equals 10 centimeters. Their resultant, π
equals negative 20π minus 16π, is acting at the point πΆ that belongs to line segment π΄π΅. Given that πΉ two equals negative 30π minus 24π determine πΉ one and the length of π΅πΆ.

In this statement, weβre told the length of line segment π΄π΅, 10 centimeters. Weβre also told the vector π
resulting from the addition of πΉ one and πΉ two. And weβre told the components of πΉ two. We want to determine πΉ one as well as the length of line segment π΅πΆ.

To start off, letβs draw a diagram of these two forces, πΉ one and πΉ two, and the line that theyβre acting on. This line weβve drawn is line segment π΄π΅, which weβre told has a length of 10 centimeters. πΉ one acts perpendicular to that line segment at point π΄. And πΉ two acts in the opposite direction at point π΅.

To solve for πΉ one, the components of that force, we can recall that π
, the resultant force, is equal to πΉ one plus πΉ two. If we wrote out this equation by the components of π
, πΉ one, and πΉ two, it would look like this. The πth component of πΉ one minus 30 the πth component of πΉ two is equal to the πth component of π
negative 20. This must mean that πΉ sub one π is equal to positive 10. Thatβs the only value that makes this sum work.

When we look at the π component, πΉ sub one π minus 24 the πth component of πΉ two is equal to negative 16 the πth component of π
. This sum implies that πΉ sub one π is equal to positive eight.

Combining these components, we can write that πΉ one equals 10π plus eight π. Those are the components of the force acting at point π΄. Next, we wanna solve for the distance of the line segment π΅πΆ, where πΆ is some distance away from π΅ such that itβs the location of the line of action of the resultant force π
.

In order to find that distance π΅πΆ, we can consider the magnitude of πΉ two to πΉ one, that is, the ratio of those two forces. If we look at the πth component of πΉ one and πΉ two, we see that the πth component of πΉ two is three times greater in magnitude than that of πΉ one, and likewise with the πth component. The πth component of πΉ two is three times greater than that of πΉ one in magnitude.

This tells us that the length of line segment π΄πΆ must be equal to three times the length of line segment π΅πΆ. Thatβs the condition we must meet in order for πΆ to be the line of action of these two forces.

Now we know that π΄πΆ is equal to π΄π΅ plus π΅πΆ. And weβve written above that that must be equal to three times π΅πΆ. So now we want to solve this equation for π΅πΆ. It simplifies to two π΅πΆ equals π΄π΅ or π΅πΆ equals π΄π΅ divided by two.

We know that π΄π΅ is equal to 10 centimeters. π΅πΆ therefore must be equal to five centimeters. Thatβs the distance from point π΅ to the line of action of these two forces.