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.
