Find the vector equation of the straight line that is parallel to the 𝑥-axis and passing through the point negative five, two.
Let’s begin by recalling the general form for the vector equation of a straight line. It’s 𝐫 equals 𝐚 plus 𝑘 times the vector 𝐝. The vector 𝐚 is the position vector of any point on the line, in other words, a point that it passes through. 𝐝 is the direction vector of a line. And 𝑘, which is often represented using alternative letters, represents a scalar. In other words, when we’re finding the vector equation of a line, we get to the line from the origin — that’s the position vector — and then we travel in multiples of our direction vector.
To see what’s really going on with our line, let’s sketch it. It’s parallel to the 𝑥-axis, and it passes through the point negative five, two. And so, it will look a little something like this. Now, we can see that it passes through the point negative five, two. But 𝐚 is a position vector of a point on the line. Essentially, that’s the vector that describes the location of our point relative to the origin.
Since the origin has coordinates zero, zero, to get from the origin to a point on the line, we travel five units to the left and two units up. That has the vector negative five, two. Note that we can represent this using angled brackets or column notation as shown. So, 𝐚 is the vector negative five, two. But what about our direction vector?
We saw that the line is parallel to the 𝑥-axis. And so, it will never travel up or down. The direction vector will, therefore, be any multiple of the vector one, zero. One, zero is the simplest vector we can choose. Essentially, it’s a unit vector. So, we can say 𝐝 is the vector one, zero. And we’re ready to write the vector equation of our line. 𝐫 is negative five, two plus 𝑘 times one, zero, where 𝐫 itself is a vector, but 𝑘 is a scalar quantity.