Feet of perpendiculars and images

What you will be able to do

• write the foot of the perpendicular from a point to an axis or to a coordinate plane;
• write the image of a point in a plane, in an axis and in the origin;
• check an image with the midpoint of the point and its image.

A shadow and a mirror

Hold a torch straight above a bird that flies at $P$. Its shadow on the floor is the point of the floor nearest to the bird. This shadow is the foot of the perpendicular from $P$ to the floor. Now put a mirror in the floor. The bird appears at the same depth below the floor. That is the image of $P$ in the plane.

Both ideas are very simple for the coordinate planes and the axes, because the perpendiculars run along the other axes.

The foot of the perpendicular

To drop a perpendicular on the $xy$-plane we move straight down. The $z$-coordinate becomes $0$, and $x$ and $y$ stay the same. The same idea works for the other planes and for the axes.

For an axis we keep the one coordinate of that axis. The other two become zero, because the perpendicular runs along the other two axes.

The image of a point

The image is as far on the other side of the plane or the axis as the point is on its own side. So the foot of the perpendicular is the midpoint of $P$ and its image. Doubling the distance from $P$ to the foot gives the image.

A plane changes one sign. An axis changes two signs. The origin changes all three.

A worked example

Take $P(3, -4, 5)$.

1. Foot on the $xy$-plane. Make $z$ zero: $(3, -4, 0)$.
2. Image in the $xy$-plane. Change the sign of $z$: $(3, -4, -5)$.
3. Foot on the $y$-axis. Keep $y$: $(0, -4, 0)$.
4. Image in the $y$-axis. Change the signs of $x$ and $z$: $(-3, -4, -5)$.
5. Check with the midpoint. The midpoint of $(3, -4, 5)$ and $(-3, -4, -5)$ is $(0, -4, 0)$, the foot. It agrees.

Practise with new numbers

Try it in Python

The program writes the feet and the images of one point and then tests the midpoint rule.

Common mistakes

• Changing too many signs. The image in a plane changes only the sign of the coordinate that is perpendicular to the plane.
• Mixing up the image and the foot. The foot is halfway. The image is twice as far.
• Changing the sign of the coordinate of the axis. The image in the $x$-axis keeps $x$ and changes $y$ and $z$.
• Thinking the foot on an axis keeps two coordinates. It keeps one. A plane keeps two.

Remember these

• Foot on a plane: set the coordinate perpendicular to it to $0$. 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 three.
• The foot is the midpoint of the point and its image.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the foot of the perpendicular from $(7, -2, 5)$ to the $yz$-plane.

2. Find the image of $(7, -2, 5)$ in the $zx$-plane.

3. Find the image of $(-3, 6, -1)$ in the $z$-axis.

4. Find the foot of the perpendicular from $(-3, 6, -1)$ to the $x$-axis.

5. A point and its image in the $xy$-plane are 14 units apart. What is the absolute value of the $z$-coordinate of the point?

References

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