UNISA
Assignment 5
MAT1503
, Question 1
1.1)
a)
Let: 𝑢
⃗ be the normal of plane U and 𝑣 be the normal of plane V.
⃗ = <λ, 5, -2λ>
𝑢
𝑣 = <-λ, 1, 2>
For the plane U & V to be orthogonal 𝑢
⃗ .𝑣=0
𝑢
⃗ .𝑣=0
<λ, 5, -2λ> . <-λ, 1, 2> = 0
-λ2 + 5 – 4λ = 0
(λ - 1)(λ + 5) = 0
Therefore
λ needs to either be -1 or 5 for the two planes to be orthogonal.
b)
Let: 𝑢
⃗ be the normal of plane U and 𝑣 be the normal of plane V.
⃗ = <λ, 5, -2λ>
𝑢
𝑣 = <-λ, 1, 2>
For the Plan U & V to be parallel 𝑢
⃗ x𝑣=0
𝑢
⃗ x𝑣=0
𝑖 𝑘𝑗
[𝜆 5 −2𝜆] = 0
𝜆 1 2
i [5 −2𝜆] - j [𝜆 −2𝜆] + k [𝜆 5] = 0
1 2 𝜆 2 𝜆 1
(10 + 2𝜆)) i – (2λ - 2λ2) j + (λ + 5λ)) k = 0
Assignment 5
MAT1503
, Question 1
1.1)
a)
Let: 𝑢
⃗ be the normal of plane U and 𝑣 be the normal of plane V.
⃗ = <λ, 5, -2λ>
𝑢
𝑣 = <-λ, 1, 2>
For the plane U & V to be orthogonal 𝑢
⃗ .𝑣=0
𝑢
⃗ .𝑣=0
<λ, 5, -2λ> . <-λ, 1, 2> = 0
-λ2 + 5 – 4λ = 0
(λ - 1)(λ + 5) = 0
Therefore
λ needs to either be -1 or 5 for the two planes to be orthogonal.
b)
Let: 𝑢
⃗ be the normal of plane U and 𝑣 be the normal of plane V.
⃗ = <λ, 5, -2λ>
𝑢
𝑣 = <-λ, 1, 2>
For the Plan U & V to be parallel 𝑢
⃗ x𝑣=0
𝑢
⃗ x𝑣=0
𝑖 𝑘𝑗
[𝜆 5 −2𝜆] = 0
𝜆 1 2
i [5 −2𝜆] - j [𝜆 −2𝜆] + k [𝜆 5] = 0
1 2 𝜆 2 𝜆 1
(10 + 2𝜆)) i – (2λ - 2λ2) j + (λ + 5λ)) k = 0
