Distance between two points (3D)

Three-dimensional space is described using three mutually perpendicular axes, $x$, $y$, and $z$, meeting at the origin $O$. Any point $P$ in space is fixed by an ordered triple of coordinates $(x,y,z)$, exactly the way a point on paper is fixed by $(x,y)$. The distance formula answers one question: given two such triples, how far apart are the points they describe? For points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$,

where: $(x_2-x_1)$, $(y_2-y_1)$, and $(z_2-z_1)$ are the differences of the corresponding coordinates of $Q$ and $P$, and $PQ$ is the straight-line distance between the two points, always taken as the positive square root. A useful special case is the distance of a point from the origin. Setting $P = O(0,0,0)$ and $Q = (x,y,z)$ in the formula above:

where: $x$, $y$, $z$ are the coordinates of the point $P$ itself, since each coordinate of the origin is zero. The formula is symmetric — it does not matter which point you call "first" and which "second," since every difference gets squared anyway. It also reduces correctly to the familiar 2-D formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ whenever the two points share a $z$-coordinate, because that term then vanishes.

Three standard uses of this formula recur across problems: checking whether three points are collinear (compute all three pairwise distances; the largest equals the sum of the other two exactly when the points lie on one line), classifying a triangle as isosceles, equilateral, or right-angled using squared distances and the converse of the Pythagorean theorem, and checking whether a point is equidistant from two given points, which is useful for locus-type questions. A fourth use, less common but still examinable, is deciding whether four points form a parallelogram, rectangle, or square by comparing the lengths of the four sides and the two diagonals, all computed with the same formula — equal opposite sides indicate a parallelogram, and equal diagonals on top of that indicate a rectangle. Working with squared distances instead of distances themselves is a habit worth building early: it avoids carrying square roots through an entire multi-step problem and only takes the root at the very end, which keeps the arithmetic cleaner and the sign-handling more transparent.

Question: Find the distance between the points $A(1,-2,3)$ and $B(4,2,-1)$, and hence check whether $A$, $B$, and the origin $O(0,0,0)$ could be the vertices of a right-angled triangle.

Step 1. Apply the distance formula to $A(1,-2,3)$ and $B(4,2,-1)$:

Since $41$ is prime, this does not simplify further; $\sqrt{41} \approx 6.4$ units.

Step 2. Find the distance of each point from the origin using $OP=\sqrt{x^2+y^2+z^2}$:

Step 3. Test the right-angle condition using squares of the three sides: $OA^2 = 14$, $OB^2 = 21$, $AB^2 = 41$. Check whether the largest squared side equals the sum of the other two: $OA^2 + OB^2 = 14 + 21 = 35 \ne 41 = AB^2$.

Step 4. Since $OA^2+OB^2 \ne AB^2$ (and no other pairing works either, as $AB^2=41$ is already the largest), the converse of the Pythagorean theorem fails, so triangle $OAB$ is not right-angled. The distance $AB=\sqrt{41}\approx6.4$ units, and $O$, $A$, $B$ do not form a right-angled triangle.

• Mishandling signs when a coordinate is already negative, e.g. computing $(2-(-2))$ as $0$ instead of $4$.
• Being inconsistent about subtraction order across the three coordinates within a single distance calculation, which does not change the squared result but causes arithmetic slips if done carelessly.
• Declaring a triangle right-angled or isosceles by eyeballing coordinates instead of actually computing and comparing all three squared distances.
• Leaving a surd unsimplified when it could be reduced (writing $\sqrt{50}$ instead of $5\sqrt2$), or trying to force-simplify a surd like $\sqrt{41}$ that is already in simplest form since $41$ is prime.

Ask the tutor: if you get stuck classifying a triangle from squared distances or want more collinearity practice, ask the AI tutor for similar 3-D coordinate problems with step-by-step checking.