Lesson 10 of 10 · 11 min

Chapter review

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15 facts

1Three mutually perpendicular axes meet at the origin O(0, 0, 0).
2Coordinate planes: XY (z = 0), YZ (x = 0), ZX (y = 0).
3The three coordinate planes divide space into 8 octants.
4Octant signs: I (+,+,+), II (−,+,+), III (−,−,+), IV (+,−,+), V (+,+,−), VI (−,+,−), VII (−,−,−), VIII (+,−,−).
5Octants V to VIII lie directly below I to IV.
6x, y and z are the distances of P from the YZ, ZX and XY-planes.
7On the x-axis: (x, 0, 0); on the y-axis: (0, y, 0); on the z-axis: (0, 0, z).
8On the YZ-plane: (0, y, z); on the XY-plane: (x, y, 0); on the ZX-plane: (x, 0, z).
9PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
10Distance from the origin: √(x² + y² + z²).
11(1, −3, 4) to (−4, 1, 2) is 3√5.
12Collinear: the longest distance equals the sum of the other two.
13Right angle: square of the longest side = sum of the squares of the other two.
14Parallelogram: opposite sides equal; a rectangle also has equal diagonals.
15Centroid = average of the three vertices.

Common traps

Saying x is the distance of P from the x-axis.

x is the distance from the YZ-plane. The distance from the x-axis is √(y² + z²).

Writing points of the ZX-plane as (x, y, 0).

The plane lacks the letter y, so y = 0 on it: (x, 0, z). (x, y, 0) is the XY-plane.

Numbering the lower octants in the wrong order.

V to VIII repeat the x, y signs of I to IV with z negative: V is (+, +, −), VI is (−, +, −), VII is (−, −, −), VIII is (+, −, −).

Dropping a sign in a coordinate difference, such as 1 − (−3) = −2.

Subtracting a negative adds: 1 − (−3) = 4. Write each difference in brackets before squaring.

Adding √14 + √56 and comparing with √126 as decimals.

Simplify first: √56 = 2√14 and √126 = 3√14, so the sum is exact.

Testing a right angle with the wrong side as hypotenuse.

Only the largest squared side can equal the sum of the other two; find it first.

Calling a parallelogram a rectangle because its opposite sides are equal.

Opposite sides equal gives a parallelogram only. Also check the diagonals: equal diagonals make it a rectangle.

5 to know

Distance between two points

PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

Pythagoras twice across the box with diagonal PQ.

Distance from the origin

OP = √(x² + y² + z²)

Points on the axes

(x, 0, 0), (0, y, 0), (0, 0, z)

x, y and z-axes.

Points on the coordinate planes

XY: (x, y, 0); YZ: (0, y, z); ZX: (x, 0, z)

The missing letter is the zero coordinate.

Centroid of a triangle

G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3)

The average of the vertices.

8 terms

Coordinate axes: Three mutually perpendicular lines through the origin: the x, y and z-axes.
Coordinate planes: The XY, YZ and ZX-planes, each fixed by a pair of axes.
Origin: The point O(0, 0, 0) where the three axes meet.
Octant: One of the eight parts into which the coordinate planes divide space.
Coordinates of a point: The ordered triplet (x, y, z) of signed distances of the point from the YZ, ZX and XY-planes.
Rectangular parallelepiped: A box with six rectangular faces; the distance formula uses one whose diagonal is PQ.
Collinear points: Points that lie on one straight line.
Centroid: The point where the medians of a triangle meet; its coordinates are the averages of the vertices.