Given that nine, one and negative eight, 𝑚 are the direction vectors of two perpendicular straight lines, determine the value of 𝑚.
So, we’re given the direction vectors of two straight lines. You might be used to seeing these written in column notation or using angled brackets. Now, these lines are actually perpendicular to one another. So, let’s recall what that actually means. Perpendicular lines meet at a right angle; they meet at an angle of 90 degrees. Now, if two lines are perpendicular and 𝑚 one and 𝑚 two represent the slope of their lines, then 𝑚 one times 𝑚 two is equal to negative one. Another way of saying this is that 𝑚 one is equal to negative one over 𝑚 two.
In other words, to find the slope of a line which is perpendicular to the line which we know the slope of, we find the negative reciprocal of that number. So, how do we find the value of the slope of the line given its direction vector? One way to think about this is rise over run. We sometimes call this change in 𝑦 over change in 𝑥. So, let’s call the line with direction vector nine, one 𝑙 sub one. For every nine units right, we move one unit up. So, rise over run for our first line must be one over nine or one-ninth.
If we call our second line 𝑙 sub two and its slope is 𝑚 sub two, we see that the negative reciprocal of 𝑚 sub one, which is a ninth, is negative nine over one or just negative nine. Now, what this tells us is that for our second line, when we move one unit to the right, we must move nine units down. The direction vector of our second line though is negative eight 𝑚. So, how far up or down do we move when we move eight units left?
Well, we can work this out by performing the opposite to finding the slope of the line. We’re going to multiply negative eight by negative nine. That gives us a value of 𝑚 of 72. Now, let’s just check this by sketching a diagram. We said for one unit right, we move nine units down. It makes a lot of sense that if we move eight units to the left, we’ll have to move some number of units up rather than down. 𝑚 is therefore equal to 72.