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

Give the Cartesian equation of the line 𝐫 = (βˆ’3, βˆ’2, βˆ’2) + 𝑑 (4, 2, 4).

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Video Transcript

Give the Cartesian equation of the line 𝐫 equals negative three, negative two, negative two plus 𝑑 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, π‘š, 𝑛 that passes through π‘₯ sub one, 𝑦 sub one, 𝑧 sub one is given by π‘₯ minus π‘₯ sub one over 𝑙 equals 𝑦 minus 𝑦 sub one over π‘š equals 𝑧 minus 𝑧 sub one over 𝑛, where 𝑙, π‘š, and 𝑛 are nonzero real numbers.

We now need to take the direction vector four, two, four to have the values of 𝑙, π‘š, and 𝑛, respectively. We can do the same and designate the coordinate π‘₯ 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 π‘₯ minus negative three over four equals 𝑦 minus negative two over two equals 𝑧 minus negative two over four. Simplifying the numerators, we have π‘₯ 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.

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