A point in the plane needs two numbers (x, y); a point in space needs a third, z, for its height.
Take the floor as the xy-plane and stand the z-axis straight up from it: drop a perpendicular from P to the floor, and its foot H is (x, y, 0) while its length is z.
The distance between two points and the equation of a sphere are the plane formulas with one z term added, so there is less new to memorize than it first seems.
Move P with the x, y and z sliders and its shadow on the floor and its distance from the origin change with it; the second figure measures from A to that same P, and in the third the r slider resizes the sphere and the right side of its equation together.
Points in Coordinate Space
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Coordinate space with the z-axis standing on the floor (the xy-plane), and the point P(x, y, z)
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4
2
👀 Reading the three coordinates
①The foot of the perpendicular from P to the floor is H(x, y, 0), and the length of PH is the z-coordinate
②The dashed lines from H to the x- and y-axes land on the x- and y-coordinates
③The dashed line from P to the z-axis carries the height z over to the point (0, 0, z)
④OH and PH are the legs of a right triangle whose hypotenuse is OP, so the distance from the origin to P is √(x² + y² + z²)
Distance Between Two Points
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From A(1, 1, 1) to B: the path that turns along x, y and z, and the straight distance d
Distance between two points
d = √((x2−x1)² + (y2−y1)² + (z2−z1)²)
The Pythagorean theorem used twice
📐 Where the formula comes from
①Going Δx in the x-direction and Δy in the y-direction from A makes a right triangle parallel to the floor, with hypotenuse √(Δx² + Δy²)
②From there, going Δz up or down reaches B, and that hypotenuse meets Δz at a right angle
③d is the hypotenuse of this second right triangle, so d² = Δx² + Δy² + Δz²
④A stays at (1, 1, 1) and B follows the point P from the first figure. Move P to (1, 1, 1) and d becomes 0
Equation of a Sphere
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The sphere formed by the points P at distance r from the center C(a, b, c)
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Equation of a sphere
(x−a)² + (y−b)² + (z−c)² = r²
Center C(a, b, c), radius r
🌐 Writing a sphere as an equation
①A sphere is the set of all points P whose distance from a fixed point C is the same value r
②Writing CP = r with the distance formula and squaring both sides gives (x − a)² + (y − b)² + (z − c)² = r²
③Expanding it gives the form x² + y² + z² + Dx + Ey + Fz + G = 0; when an equation arrives in that form, complete the square in each variable to recover the center and radius
④In the figure the center stays put and only r changes. Doubling r makes the right side r² four times as large
Division Points and the Centroid
Internal division point
P = (nx1+mx2m+n, ny1+my2m+n, nz1+mz2m+n)
The point dividing segment AB in the ratio m : n
Centroid of a triangle
G = (x1+x2+x33, y1+y2+y33, z1+z2+z33)
The average of the three vertices
💡 The plane formula plus one line for z
①The division point applies the same ratio m : n to the x-, y- and z-coordinates separately
②It is the plane formula with one more line for z
③The midpoint is the case m = n = 1, so each coordinate is an average
④The centroid averages the x-, y- and z-coordinates of the three vertices separately
Worked Examples
Example 1
Find the distance between the two points A(1, 2, 3) and B(4, 6, 3).
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Subtract coordinate by coordinate, square each difference and add.
AB = √((4−1)² + (6−2)² + (3−3)²)
2
Take the square root of the sum, 25.
AB = √(9 + 16 + 0) = √25 = 5
▸ 5
The two z-coordinates are equal, so Δz = 0 and the distance is the same as in the flat plane at that height.
Example 2
Find the center and radius of the sphere x² + y² + z² − 2x + 4y − 6z + 5 = 0.
Match it with the standard form and read off the center and radius.
(x−1)² + (y+2)² + (z−3)² = 1+4+9−5 = 9
▸ center (1, −2, 3), radius 3
The right side 9 is r², so the radius is 3. If completing the square left a negative right side, no point would satisfy the equation.
Summary
Distance from the origin
d = √(x² + y² + z²)
From the origin O to the point P(x, y, z)
2023 KICE mock exam Math (Geometry) type, adapted
Points A(2, 0, 1) and B(0, 4, 3) are the ends of a diameter of a sphere. What is the center of the sphere?
①(1, 2, 2)
②(2, 2, 1)
③(1, 2, 1)
④(2, 4, 2)
⑤(1, 4, 2)
▸ ① (1, 2, 2)
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The center of the sphere is the midpoint of the diameter AB.
2
Average the x-, y- and z-coordinates of the two points separately.
((2+0)/2, (0+4)/2, (1+3)/2) = (1, 2, 2)
🎯 Exam Points
①The distance formula is the plane formula plus one (z₂ − z₁)² term
②For a sphere given in general form, complete the square to read off the center and radius. A right side of 0 gives a single point, and a negative one gives no figure at all
③Division points, midpoints and centroids are worked out one coordinate at a time
④For a symmetric point, which coordinates change sign depends on whether the symmetry is about an axis, a coordinate plane or the origin