MAT2615 Assignment 1 (COMPLETE ANSWERS) 2025 - DUE 15 May 2025; 100% TRUSTED Complete, trusted solutions and explanations..

1. Equation of the Plane Containing Two Lines Given parametric equations:


ℓ1:(𝑥,𝑦,𝑧)

(1,0,0)+𝑡(1,0,1),𝑡∈𝑅ℓ1 :
(x,y,z)=(1,0,0)+t(1,0,1),t∈R ℓ 2 : ( 𝑥 , 𝑦 , 𝑧 )
( 1 , 0 , − 1 ) + 𝑡 ( 0 , 1 , 1 ) , 𝑡 ∈ 𝑅 ℓ 2 :(x,y,z)=(1,0,−1)+t(0,1,1),t∈R Step 1: Find Two
Direction Vectors


𝑑1

(1,0,1),𝑑2
( 0 , 1 , 1 ) d 1 =(1,0,1),d 2 =(0,1,1) Step 2: Compute Normal Vector (Cross Product)


𝑛

𝑑1×𝑑2

∣ 𝑖 𝑗 𝑘 1 0 1 0 1 1 ∣ n=d 1 ×d 2

i10

j01

k11



Expanding along the first row:

, 𝑛
( − 1 , − 1 , 1 ) n=(−1,−1,1) Step 3: Find Plane Equation Using point ( 1 , 0 , 0 ) (1,0,0),


−(𝑥−1)−𝑦+𝑧

0 −(x−1)−y+z=0 𝑥 + 𝑦 − 𝑧
1 x+y−z=1 Final Answer:


𝑥+𝑦−𝑧
1 x+y−z=1 2. Find Parametric Equation for the Intersection of Two Planes Given:


3𝑥+2𝑦−𝑧−4

0 3x+2y−z−4=0 − 𝑥 − 2 𝑦 + 2 𝑧
0 −x−2y+2z=0 Step 1: Express Variables in Terms of Parameter 𝑡 t Solve for 𝑧 z from
the second equation:


𝑧
𝑥 + 2 𝑦 2 z= 2 x+2y

Substituting into the first equation: