---
title: "Parametric line in 3D: point and plane intersection"
subject: "Computer Aided Design"
chapter: 3
key: "semester_4/computer_aided_design:parametric-line-segment-plane"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/parametric-line-segment-plane
site: BCT Engineering
---

# Parametric line in 3D: point and plane intersection

Computer Aided Design · Chapter 3: Mathematical Representation of Curves

- Asked in: year not given, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/parametric-line-segment-plane
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-3

## Question

A line segment joins P1(1, 2, 3) and P2(5, -2, 9). (a) Write its parametric equation. (b) Find its length and direction cosines. (c) Find the point at u = 0.25. (d) Find the point where the line meets the plane z = 6.

## Short answer

Parametric equation is P(u)=(1+4u, 2-4u, 3+6u), 0<=u<=1. Length is 8.246; direction cosines are (0.485, -0.485, 0.728). At u=0.25 the point is (2, 1, 4.5). Intersection with z=6 occurs at u=0.5, giving point (3, 0, 6).

## Answer

#### (a) Parametric equation
$$
\mathbf P(u)=\mathbf P_1+u(\mathbf P_2-\mathbf P_1),\qquad 0\le u\le1
$$

$\mathbf P_2-\mathbf P_1=(4,\,-4,\,6)$, so

$$
x=1+4u,\qquad y=2-4u,\qquad z=3+6u
$$

#### (b) Length and direction cosines
$$
L=\sqrt{4^2+(-4)^2+6^2}=\sqrt{68}=8.246
$$

$$
l=\frac{4}{8.246}=0.485,\quad m=\frac{-4}{8.246}=-0.485,\quad n=\frac{6}{8.246}=0.728
$$

Check: $l^2+m^2+n^2=0.235+0.235+0.529=1.0$.

#### (c) Point at u = 0.25
$$
x=1+1=2,\quad y=2-1=1,\quad z=3+1.5=4.5
$$

Point $(2,\ 1,\ 4.5)$.

#### (d) Intersection with z = 6
$3+6u=6 \Rightarrow u=0.5$, which lies in $0\le u\le1$, so the segment does cut the plane.

$$
x=1+4(0.5)=3,\quad y=2-4(0.5)=0
$$

**Answer: L = 8.246; direction cosines (0.485, -0.485, 0.728); P(0.25) = (2, 1, 4.5); intersection with z = 6 at (3, 0, 6).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is the parametric equation of the segment?
   A. P = P2 + u(P1 + P2)
   B. P = u P1 + u P2
   C. P = P1 + (P2 - P1)/u
   D. P = P1 + u(P2 - P1), 0 ≤ u ≤ 1
   Answer: D. A segment is P1 plus u times the direction P2 - P1. (Key point: Line equation)

2. What is the length of the segment from (1,2,3) to (5,-2,9)?
   A. 8.246
   B. 6.0
   C. 10.0
   D. 68
   Answer: A. sqrt(16 + 16 + 36) = sqrt(68) = 8.246. (Key point: Length)

3. What is the point at u = 0.25?
   A. (1.25, 1.75, 3.25)
   B. (2, 1, 4.5)
   C. (3, 0, 6)
   D. (2, 1, 3.5)
   Answer: B. x = 1+4(0.25)=2, y = 2-1=1, z = 3+1.5=4.5. (Key point: Point at u)

4. How is the intersection with z = 6 found?
   A. Set 1 + 4u = 6
   B. Use direction cosines only
   C. Set 3 + 6u = 6, so u = 0.5, then find x and y
   D. Set u = 1
   Answer: C. Solve z(u) = 6 for u = 0.5 (within 0 to 1), giving (3, 0, 6). (Key point: Plane intersection)

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