Coordinates of a Point in Space: Quick Revision

A brief note on Coordinates of a Point in Space. Read it before a test. Try each question at the end before you open its answer.

Axes, planes and octants

• Axes: three perpendicular lines through the origin $O(0, 0, 0)$, in the right-handed order $x$, $y$, $z$.
• Coordinate planes: $xy$-plane ($z = 0$), $yz$-plane ($x = 0$), $zx$-plane ($y = 0$).
• On an axis: two coordinates are zero. In a plane: one coordinate is zero.
• Octants: eight, decided by the signs of $(x, y, z)$. A point with a zero coordinate is in no octant.

Feet and images

• Foot on a plane: make the coordinate perpendicular to the plane zero. Foot on an axis: keep only that axis's coordinate.
• Image in a plane: change one sign. Image in an axis: change two signs. Image in the origin: change all three.
• Midpoint rule: the foot is the midpoint of the point and its image.

Distances of $P(x, y, z)$

• From the origin: $\sqrt{x^2 + y^2 + z^2}$.
• From the axes: $\sqrt{y^2 + z^2}$, $\sqrt{x^2 + z^2}$, $\sqrt{x^2 + y^2}$.
• From the planes: $|z|$, $|x|$, $|y|$. From the plane $z = k$: $|z - k|$.

A worked example

Find the image of $P(4, -3, 12)$ in the $y$-axis and the distance of $P$ from the $z$-axis.

1. Image. The $y$-axis keeps $y$ and changes the other two signs: $(-4, -3, -12)$.
2. Distance. $\sqrt{4^2 + (-3)^2} = 5$.

Practice with new numbers

A program to try

The program works out the main results for one point.

Mistakes to avoid

• Swapping the order of $x$, $y$, $z$.
• Changing too many signs in an image. A plane changes one, an axis changes two.
• Using all three squares for the distance from an axis. Leave out the coordinate along the axis.
• Giving an octant to a point with a zero coordinate.

Check yourself

1. Write the foot of the perpendicular from $(5, -2, 8)$ to the $zx$-plane.

2. Write the image of $(5, -2, 8)$ in the $z$-axis.

3. Find the distance of $(2, 3, 6)$ from the origin.

4. In which octant is $(-1, 3, -7)$?

5. Find the distance of $(3, 5, 4)$ from the plane $z = 9$.

Sources

• NCERT, Mathematics Textbook for Class XI, the chapter Introduction to Three Dimensional Geometry.
• OpenStax, Calculus Volume 3, Chapter 2, section 2.2 on vectors in three dimensions.