Distance Calculations

Calculus 3 · Axiom Academy

LESSON Distance Calculations in 3D Point to point, point to line, point to plane — every distance in space is a perpendicular measurement. 1. The Distance Between Two Points Two points in space determine a box. Take P(1, 2, 3) and Q(10, 14, 23) : to travel from P to Q you go one way, another, and straight up. The straight-line distance d is the space diagonal of that box — and two rounds of the Pythagorean theorem unlock it: once across the floor, once up the wall. Pythagoras applied twice — the floor diagonal first, then the climb Before tackling lines and planes, we need one tool. The projection of a vector onto a vector tells us how much of goes in the direction of — it is the shadow that casts on 's line when the light falls perpendicular to it. Watch the shadow accumulate, point by point: 3. Distance from a Point to a Line A line passes through P_0 with direction vector , and a point P sits off the line. The shortest path from P to the line meets it at a right angle. To find its length: Form the vector from the line to the point. Project it onto the line: is the part that runs along the line. Subtract. What remains is perpendicular to the line — its length is the distance D . For the line through P_0(1,1,1) with direction and the point P(3,4,5) : here , so . Subtracting leaves , whose length is . 4. A Shortcut: The Cross Product

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