Question Video: Finding the Cartesian Form of the Equation of the Straight Line That Passes through a Given Point and Makes Equal Angles with the Coordinate Axes Mathematics

Find the Cartesian form of the equation of the straight line passing through the point (−4, 1, 2) and makes equal angles with the coordinates axes.

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Video Transcript

Find the Cartesian form of the equation of the straight line passing through the point negative four, one, two and makes equal angles with the coordinates axes.

We begin by recalling that the Cartesian form of the equation of a straight line is 𝑥 minus 𝑥 sub zero over 𝑙 is equal to 𝑦 minus 𝑦 sub zero over 𝑚, which is equal to 𝑧 minus 𝑧 sub zero over 𝑛, where the line has direction vector 𝑙, 𝑚, 𝑛 and passes through the point with coordinates 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero.

We are told in this question that our line passes through the point with coordinates negative four, one, two. This means that 𝑥 sub zero is equal to negative four, 𝑦 sub zero is equal to one, and 𝑧 sub zero equals two. At first glance, it might not appear that we know anything about the direction vector of the line. However, we are told that the line makes equal angles with the coordinate axes.

If we consider the two-dimensional 𝑥𝑦-coordinate plane, then a line that makes equal angles with the two coordinate axes will have slope or gradient one. This means that for every one unit we move in the 𝑥-direction, we’ll move one unit in the 𝑦-direction. This line has a direction vector one, one. We can extend this to three dimensions such that the direction vector that makes equal angles with the 𝑥-, 𝑦-, and 𝑧-axis is one, one, one. The values of 𝑙, 𝑚, and 𝑛 are therefore all equal to one. Subtracting negative four from 𝑥 is the same as adding four to 𝑥. So our first expression is 𝑥 plus four over one. This is equal to 𝑦 minus one over one and 𝑧 minus two over one.

The Cartesian form of the equation of the straight line that passes through the point negative four, one, two and makes equal angles with the coordinate axes is 𝑥 plus four over one is equal to 𝑦 minus one over one, which is equal to 𝑧 minus two over one.

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