Functions of two variables and partial derivatives practice question 5
Which statement about the point in three dimensions is correct?
The plane is every point whose -coordinate is 5, whatever and are. Which coordinate plane never changes either?
The -plane is , the -plane is , and the -plane is . The distance from to each is the size of the coordinate that plane sets to 0.
Theory UBC Math 100 §14.1Key results
Worked solution
Strategy
Each coordinate plane is where one coordinate is 0, and the planes x = a, y = b, z = c are parallel copies of them. The distance from a point to a coordinate plane is the size of the coordinate that plane sets to 0.
Working
Every point with y-coordinate 5, P included. Like the xz-plane, y = 0, it holds y fixed and lets x and z vary, so it is parallel to the xz-plane, 5 units from it.
Not 5: the y-coordinate measures the distance to the xz-plane.
P lies on it, but it holds z fixed, so it is parallel to the xy-plane, z = 0, not the yz-plane.
P is 3 units below the xy-plane.
Answer
The answer is P lies on the plane y = 5, which is parallel to the xz-plane..
Check
The points (0, 5, 0), (1, 5, 0) and (0, 5, 1) all lie on y = 5, and moving between them changes only x or z, as moving within the xz-plane does.
Where this goes wrong
Matching a plane to the letters in its name the wrong way. The plane y = 5 holds y fixed, so it is parallel to the coordinate plane where y is also fixed, y = 0, which is the xz-plane: the plane named after the two letters that still vary.
The idea behind it
The functions of two variables and partial derivatives lesson, a figure to move, in a few steps
Leans on
Distance between two points, from Functions, graphs and inverses
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