Finding Line Through P(1,2,3) and Q(4,-1,2)

Calculus 3 · Axiom Academy

EXAMPLE Finding the Line Through P(1,2,3) and Q(4,-1,2) Build the direction vector, write the parametric equations, and convert to symmetric form. Find the equation of the line through the points P(1, 2, 3) and Q(4, -1, 2) in . Give both the parametric form and the symmetric form, and confirm that both P and Q lie on the line. Two points P and Q determine a unique line. The direction vector points from P toward Q and gives the line its heading. Nice work! You built the equation of a line in 3D from two points and confirmed both points lie on it. Direction vector: subtract corresponding coordinates, — it points from P toward Q . Parametric form: anchor at a known point and step along the direction: . Symmetric form: solve each parametric equation for t and set them equal (valid only when every component of is non-zero). Verify the endpoints: t = 0 returns P and t = 1 returns Q , so both points sit on the line. These forms are the workhorses of 3D geometry in Calculus 3 — used for tangent lines to curves, intersections of planes, and setting up line integrals.

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