Find the vector equation of the straight line whose slope is negative eight over three and passes through the point four, negative nine.
We can say that the line is made up of two pieces, a point and direction. We can use a vector equation to display the point and its direction. The vector 𝑟 is made up of the initial point plus a nonzero vector that is parallel to the line. But we also include a constant 𝑘 before this vector because any scalar of this vector that is parallel to the line will work.
We’ll use the given information to write an equation in this format. The vector 𝑟 equals a point. We’re given the first point, four, negative nine. So we’ll say our vector 𝑟 is equal to four, negative nine plus some 𝑘 values, some multiple of a parallel vector. Now, here is where it’s not exactly clear what this vector should be.
Okay, let’s imagine our initial point, point four, negative nine, to be here. We know that this point falls on a line whose slope is negative eight, three. So we move down eight and right three. And we have a line that looks like this. This pink vector also represents a slope of negative eight-thirds. It’s made up of the components three and negative eight. It’s also parallel to our original line. And we’ll substitute that value in to find our vector equation.
The vector equation for this straight line is this. The vector 𝑟 equals four, negative nine plus 𝑘 times three, negative eight. Remember that this constant 𝑘 stays here because we’re saying that any multiple of this vector would be correct.