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Question Video: Finding the Cartesian Equation of a Line through a Given Point and with a Given Direction Vector Mathematics

Give the Cartesian equation of the line through point (−2, 5, 2) and with direction vector <3, −5, −4>.

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Video Transcript

Give the Cartesian equation of the line through point negative two, five, two and with direction vector three, negative five, negative four.

We know that the Cartesian equation of a line is written in the form 𝑥 minus 𝑥 sub one over 𝐥 is equal to 𝑦 minus 𝑦 sub one over 𝐦, which is equal to 𝑧 minus 𝑧 sub one over 𝐧, where the line has direction vector 𝐥, 𝐦, 𝐧. And the line passes through the point with coordinates 𝑥 sub one, 𝑦 sub one, 𝑧 sub one, where 𝐥, 𝐦, and 𝐧 must be nonzero real numbers.

In this question, we are told that the direction vector is three, negative five, negative four. This means that 𝐥 equals three, 𝐦 equals negative five, and 𝐧 equals negative four. We are also told that the line passes through the point with coordinates negative two, five, two. These are the values of 𝑥 sub one, 𝑦 sub one, and 𝑧 sub one, respectively.

Substituting these values into the general equation, we have the following. 𝑥 minus negative two over three is equal to 𝑦 minus five over negative five, which is equal to 𝑧 minus two over negative four. Subtracting negative two is the same as adding two. The Cartesian equation of the line through the point negative two, five, two and with direction vector three, negative five, negative four is 𝑥 plus two over three is equal to 𝑦 minus five over negative five, which is equal to 𝑧 minus two over negative four.

In order to prevent a negative number on the denominator, we could rewrite the second expression as five minus 𝑦 over five and the third expression as two minus 𝑧 over four.

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