---
title: "Distance of point from plane parallel to line"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:distance-point-from-plane-parallel"
source: https://www.bctengineer.com/q/semester_2/mathematics/distance-point-from-plane-parallel
site: BCT Engineering
---

# Distance of point from plane parallel to line

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2072 Ashwin, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/distance-point-from-plane-parallel
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the distance of the point (1,−2,3) from the plane x−y+z = 5 measured parallel to the line x/2 = y/3 = z/(−6).

## Short answer

Line through (1,-2,3) parallel to (2,3,-6): (1+2t, -2+3t, 3-6t). Intersection with x-y+z=5 gives t=1/7. Distance = |t|√(2²+3²+(-6)²) = (1/7)×7 = 1 unit.

## Answer

**Given.** Point P(1, −2, 3), plane x − y + z = 5, and the direction of the line x/2 = y/3 = z/(−6), which is (2, 3, −6).

#### Method
Draw the line from P parallel to (2, 3, −6) and find where it meets the plane. The distance from P to that point is the required distance.

A point on the line through P: (1 + 2t, −2 + 3t, 3 − 6t).

#### Meeting the plane
```text
(1 + 2t) − (−2 + 3t) + (3 − 6t) = 5
1 + 2t + 2 − 3t + 3 − 6t = 5
6 − 7t = 5
t = 1/7
```

The meeting point is Q(1 + 2/7, −2 + 3/7, 3 − 6/7) = (9/7, −11/7, 15/7).

#### Distance
```text
PQ = |t| √(2² + 3² + (−6)²) = (1/7)(7) = 1
```

Direct check: PQ = (2/7, 3/7, −6/7), and √(4 + 9 + 36)/7 = 7/7 = 1.

**Answer: the distance is 1 unit.**

## Check yourself

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

1. What is the distance measured parallel to a line?
   A. The distance from P to the point where the line through P parallel to that direction meets the plane
   B. The perpendicular distance from P to the plane
   C. The distance from P to the line
   D. The distance from the origin to the plane
   Answer: A. The path from P follows the given direction until it hits the plane. (Key point: Meaning of the distance)

2. What is t for the meeting point?
   A. t = -1/7
   B. t = 1/7
   C. t = 7
   D. t = 1
   Answer: B. 6 - 7t = 5. (Key point: Parameter value)

3. How is the distance found?
   A. |t| = 1/7
   B. t x (2 + 3 - 6) = 1/7
   C. |t| x sqrt(2^2 + 3^2 + 6^2) = 1
   D. |t| x 3 = 3/7
   Answer: C. Each step of t moves 7 units, so (1/7)(7) = 1. (Key point: Distance)

## Related in Mathematics

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