Find the parametric equations of the straight line that passes through the point negative nine, eight with direction vector four, negative seven.
The parametric equations of a straight line means that we want to express both 𝑥 and 𝑦 in terms of a parameter, which we’ll call 𝑘. 𝑥 will be some function 𝑓 of 𝑘. And 𝑦 will be another function 𝑔 of 𝑘. We’re told the coordinates of a point that this straight line passes through and its direction vector. So we’ll begin by writing the vector equation of the line.
Every point on the line can be reached by starting at this point with coordinates negative nine, eight and then travelling some multiple of the direction vector. So the vector equation of the line is 𝑟 is equal to negative nine, eight plus 𝑘 lots of four, negative seven. To convert this to parametric equations, we can express the general point on the line 𝑟 as the point 𝑥, 𝑦.
To find the parametric equations, we then just need to equate component parts of the vectors. To find the equation for 𝑥, we equate the first component of each vector. This gives 𝑥 is equal to negative nine plus 𝑘 multiplied by four, four 𝑘. To find the equation for 𝑦, we equate the second component of each vector. This gives 𝑦 is equal to eight plus 𝑘 multiplied by negative seven, so minus seven 𝑘. The parametric equations of this straight line are 𝑥 equals negative nine plus four 𝑘, 𝑦 equals eight minus seven 𝑘.