Distance and Section Formulae in Space: Quick Revision

A brief note on Distance and Section Formulae in Space. Read it before a test. Try each question at the end before you open its answer.

The distance

• Distance: $PQ^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2$, and $PQ$ is the square root.
• Equidistant point on an axis: write $(t, 0, 0)$, set the squared distances equal. The $t^2$ terms cancel.
• Triangle types: compare squared sides. $a^2 + b^2 = c^2$ means a right angle.

Section formulae

• Internal ratio $m : n$: $\dfrac{mB + nA}{m + n}$, coordinate by coordinate. Midpoint: $m = n = 1$.
• External ratio $m : n$: $\dfrac{mB - nA}{m - n}$, with $m \ne n$.
• Ratio from a coordinate: $\lambda = \dfrac{x - x_1}{x_2 - x}$. All three coordinates must agree. A negative $\lambda$ means external.
• A coordinate plane such as $x = 0$ divides $AB$ in the ratio $-x_1 : x_2$.

Centroids, lines and shapes

• Centroid of a triangle: the mean of the 3 corners. It cuts a median in the ratio $2 : 1$.
• Centroid of a tetrahedron: the mean of the 4 corners. It cuts a line to the opposite face in the ratio $3 : 1$.
• Collinear: $AB + BC = AC$, or one ratio from all three coordinates.
• Parallelogram $ABCD$: $A + C = B + D$. Rectangle: also $AC = BD$. Rhombus: also $AB = BC$.

A worked example

Find the point that divides $A(1, 2, 3)$ and $B(7, -4, 9)$ in the ratio $1 : 2$, and the distance $AP$.

1. The point. $P = \dfrac{1 \times B + 2 \times A}{3} = (3, 0, 5)$.
2. The distance. $AP = \sqrt{4 + 4 + 4} = 2\sqrt{3}$, which is one third of $AB = 6\sqrt{3}$.

Practice with new numbers

A program to try

The program checks the main formulae on small examples.

Mistakes to avoid

• Swapping the weights. The weight $m$ goes with $B$, the far end.
• Using $m + n$ for an external division. Use $m - n$.
• Dividing by 3 or 4 wrongly. A triangle has 3 corners and a tetrahedron has 4.
• Reading a shape from a picture. Compare squared sides and diagonals.

Check yourself

1. Find the midpoint of $(2, -4, 6)$ and $(8, 0, -2)$.

2. Find the distance between $(1, 2, 3)$ and $(4, 6, 15)$.

3. Find the point that divides $(0, 0, 0)$ and $(6, 9, 12)$ internally in the ratio $1 : 2$.

4. Find the centroid of the triangle $(0, 0, 0)$, $(6, 0, 0)$, $(0, 9, 3)$.

5. The plane $y = 0$ cuts the join of $(1, 3, 2)$ and $(5, -1, 6)$. In what ratio?

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.