--- title: "Lines intersect: point and plane through them" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:lines-intersect-point-plane-through" source: https://www.bctengineer.com/q/semester_2/mathematics/lines-intersect-point-plane-through site: BCT Engineering --- # Lines intersect: point and plane through them Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2067 Chaitra, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/lines-intersect-point-plane-through - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Prove that the lines (x−1)/2 = (y+1)/(−3) = (z+10)/8 and (x−4)/1 = (y+3)/(−4) = (z+1)/7 intersect. Also find their point of intersection and plane through them. ## Short answer Equating parametric forms gives t=2, s=1, intersection point (5,-7,6). The plane containing both lines has normal (11,-6,-5) and equation 11x - 6y - 5z = 67. ## Answer Two lines intersect if a common point exists; then they are coplanar. Line 1: (x−1)/2 = (y+1)/(−3) = (z+10)/8 = t, so (1 + 2t, −1 − 3t, −10 + 8t). Line 2: (x−4)/1 = (y+3)/(−4) = (z+1)/7 = s, so (4 + s, −3 − 4s, −1 + 7s). #### Intersection Equate the coordinates: ```text 1 + 2t = 4 + s → 2t − s = 3 ... (i) −1 − 3t = −3 − 4s → −3t + 4s = −2 ... (ii) −10 + 8t = −1 + 7s → 8t − 7s = 9 ... (iii) ``` From (i), s = 2t − 3. Put in (ii): −3t + 8t − 12 = −2, so t = 2 and s = 1. Check (iii): 8(2) − 7(1) = 9 ✓. All three equations are satisfied, so the lines intersect. **Point of intersection:** t = 2 gives (5, −7, 6). (s = 1 gives (5, −7, 6) as well.) #### Plane through them Normal = d₁ × d₂ with d₁ = (2, −3, 8), d₂ = (1, −4, 7): ```text n = ((−3)(7) − 8(−4), 8(1) − 2(7), 2(−4) − (−3)(1)) = (−21 + 32, 8 − 14, −8 + 3) = (11, −6, −5) ``` Plane through (5, −7, 6): 11(x − 5) − 6(y + 7) − 5(z − 6) = 0 → 11x − 6y − 5z − 55 − 42 + 30 = 0. **Answer: the lines meet at (5, −7, 6) and lie in the plane 11x − 6y − 5z = 67.** Check: the point (1, −1, −10) of line 1: 11 + 6 + 50 = 67 ✓. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is it shown the lines intersect? A. Check d1 . d2 = 0 B. Equate coordinates with parameters t and s and check all three equations hold C. Check the lines are parallel D. Check d1 = d2 Answer: B. A common point exists only if one (t, s) satisfies all three. (Key point: Intersection method) 2. Solving the first two equations gives: A. t = 1, s = 2 B. t = 2, s = -1 C. t = 2, s = 1 D. t = 3, s = 3 Answer: C. s = 2t - 3 into -3t + 4s = -2 gives t = 2. (Key point: Parameters) 3. The point of intersection is: A. (5, -7, 6) B. (1, -1, -10) C. (5, 7, 6) D. (4, -3, -1) Answer: A. t = 2: (1+4, -1-6, -10+16). (Key point: Point) 4. The plane through them is: A. 11x - 6y - 5z = -67 B. 11x - 6y - 5z = 67 C. 11x + 6y - 5z = 67 D. x - y - z = 67 Answer: B. Normal d1 x d2 = (11, -6, -5) through (5, -7, 6). (Key point: Plane) ## Related in Mathematics - [Coplanar lines, intersection point and plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-intersection-point-plane) — Mathematics, chapter 3, asked 2× - [Plane through intersection, perpendicular to plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aplane-through-intersection-perpendicular-plane) — Mathematics, chapter 3 - [Coplanar lines with point of intersection](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-point-intersection-2) — Mathematics, chapter 3, asked 3× - [Coplanar lines and point of intersection](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-point-intersection) — Mathematics, chapter 3, asked 2× - [Determinant form of plane through point and line](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adeterminant-form-plane-through-point) — Mathematics, chapter 3 - [Sphere through circle of two spheres, centre on a plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-through-circle-two-spheres) — Mathematics, chapter 3 ## Related in other subjects - [Parametric line in 3D: point and plane intersection](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Aparametric-line-segment-plane) — Computer Aided Design, chapter 3 - [Intersection versus resection in plane tabling](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aintersection-vs-resection-plane-table) — Surveying I, chapter 6 - [Plane through three points and distance](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Aplane-surface-three-points) — Computer Aided Design, chapter 4 - [Intersection vs resection, three point method](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aintersection-vs-resection-three-point) — Surveying II, chapter 5