Question Video: Converting the given Form of an Equation of a Straight Line to Cartesian Form Mathematics

Video Transcript

Give the Cartesian equation of the line 𝐫 equals negative three, negative two, negative two plus 𝑡 four, two, four.

In this question, we’re given this equation in vector form. Negative three, negative two, negative two is the position vector of a given point, and four, two, four is the direction vector. In order to change the equation in vector form into an equation in Cartesian form, there is a formula we can apply. The equation of a line with direction vector 𝐯 equals 𝑙, 𝑚, 𝑛 that passes through 𝑥 sub one, 𝑦 sub one, 𝑧 sub one is given by 𝑥 minus 𝑥 sub one over 𝑙 equals 𝑦 minus 𝑦 sub one over 𝑚 equals 𝑧 minus 𝑧 sub one over 𝑛, where 𝑙, 𝑚, and 𝑛 are nonzero real numbers.

We now need to take the direction vector four, two, four to have the values of 𝑙, 𝑚, and 𝑛, respectively. We can do the same and designate the coordinate 𝑥 sub one, 𝑦 sub one, 𝑧 sub one with the values negative three, negative two, negative two. Plugging these values into the formula, we have 𝑥 minus negative three over four equals 𝑦 minus negative two over two equals 𝑧 minus negative two over four. Simplifying the numerators, we have 𝑥 plus three over four equals 𝑦 plus two over two equals 𝑧 plus two over four. And that’s the answer for the Cartesian equation of the given line.