Exam (elaborations)
MAT1503 Assignment 8 with Complete Solutions 2023
- Course
- MAT1503
- Institution
- University Of South Africa
MAT1503 Assignment 8 with Complete Solutions 2023
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MAT1503 Assignment 8 withComplete Solutions 2023
QUESTION 1
QUESTION 1.1
𝑈: 𝜆𝑥 + 5𝑦 − 2𝜆𝑧 − 3 = 0
𝑉: − 𝜆𝑥 + 𝑦 + 2𝑧 + 1 = 0
𝐿𝑒𝑡 ∶ 𝑛
⃗⃗⃗⃗1→ = 𝑛𝑜𝑟𝑚𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑈 𝑎𝑛𝑑 𝑛
⃗⃗⃗⃗2→ = 𝑛𝑜𝑟𝑚𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑉
⃗⃗⃗⃗1→ = (𝜆, 5, −2𝜆) 𝑎𝑛𝑑 𝑛
𝑛 ⃗⃗⃗⃗2→ = (−𝜆, 1,2)
a).
𝐼𝑓 𝑈 𝑎𝑛𝑑 𝑉 𝑎𝑟𝑒 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙 𝑡ℎ𝑒𝑛 , 𝑛
⃗⃗⃗⃗1→ ∙ 𝑛
⃗⃗⃗⃗2→ = 0
𝑛
⃗⃗⃗⃗1→ ∙ 𝑛
⃗⃗⃗⃗2→ = 0
(𝜆, 5, −2𝜆) ∙ (−𝜆, 1,2) = 0
−𝜆2 + 5 − 4𝜆 = 0
𝜆2 + 4𝜆 − 5 = 0
(𝜆 − 1)(𝜆 + 5) = 0
𝜆 = 1 𝑜𝑟 𝜆 = −5
b).
𝐼𝑓 𝑈 𝑎𝑛𝑑 𝑉 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑖𝑓 𝑛
⃗⃗⃗⃗1→ × 𝑛
⃗⃗⃗⃗2→ = 0
𝑖 𝑗 𝑘
𝑛
⃗⃗⃗⃗1→ × 𝑛
⃗⃗⃗⃗2→ = | 𝜆 5 −2𝜆|
, −𝜆 1 2
5 −2𝜆 𝜆 −2𝜆 𝜆 5
= 𝑖| |−𝑗| |+𝑘| |
1 2 −𝜆 2 −𝜆 1
= (10 + 2𝜆)𝑖 − (2𝜆 − 2𝜆2)𝑗 + (𝜆 + 5𝜆)𝑘
= (10 + 2𝜆)𝑖 − (2𝜆 − 2𝜆2)𝑗 + (6𝜆)𝑘
𝑛 ⃗⃗⃗⃗2→ = 〈10 + 2𝜆 ,2𝜆2 − 2𝜆 ,6𝜆〉
⃗⃗⃗⃗1→ × 𝑛
𝑛
⃗⃗⃗⃗1→ × 𝑛
⃗⃗⃗⃗2→ = 0
〈10 + 2𝜆 ,2𝜆2 − 2𝜆 ,6𝜆〉 = 〈0,0,0〉
10 + 2𝜆 = 0 ⟾ 𝜆 = −5
2𝜆2 − 2𝜆 = 0 ⟾ 𝜆 = 0 𝑜𝑟 𝜆 = 1
6𝜆 = 0 ⟾𝜆=0
𝑊𝑒 𝑔𝑒𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝜆 𝑚𝑒𝑎𝑛𝑖𝑛𝑔, 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝜆 𝑤ℎ𝑒𝑛 𝑈 𝑎𝑛𝑑 𝑉 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
QUESTION 1.2
𝐿𝑒𝑡: 𝑉 𝑏𝑒 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑡ℎ𝑎𝑡 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛
𝑆𝑖𝑛𝑐𝑒 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑉 𝑖𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 ∶ −𝑥 + 3𝑦 − 2𝑧 = 6, 𝑡ℎ𝑒𝑦 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑛𝑜𝑟𝑚𝑎𝑙
𝑛⃗→ = 𝑛𝑜𝑟𝑚𝑎𝑙 𝑜𝑓 𝑉
𝑛⃗→ = 〈−1,3, −2〉
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑙𝑎𝑛𝑒 ∶ 𝑟 ∙ 𝑛⃗→ = 𝑟𝑜 ∙ 𝑛⃗→
𝑟 = 〈𝑥, 𝑦, 𝑧〉
𝑟𝑜 = 〈0,0,0〉
𝑛⃗→ = 〈−1,3, −2〉
𝑟 ∙ 𝑛⃗→ = 𝑟𝑜 ∙ 𝑛⃗→
〈𝑥, 𝑦, 𝑧〉 ∙ 〈−1,3, −2〉 = 〈0,0,0〉 ∙ 〈−1,3, −2〉
−𝑥 + 3𝑦 − 2𝑧 = 0
, QUESTION 1.3
𝐿𝑒𝑡: 𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑜𝑖𝑛𝑡 (𝑥1, 𝑦1, 𝑧1) 𝑎𝑛𝑑 𝑝𝑙𝑎𝑛𝑒 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0
|𝑎𝑥1 + 𝑏𝑦1 + 𝑐𝑧1 + 𝑑|
𝑑=
√(𝑎)2 + (𝑏)2 + (𝑐)2
(𝑥1, 𝑦1, 𝑧1) = (−1, −2,0)
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0 = 3𝑥 − 𝑦 + 4𝑧 + 2
∴ 𝑎 = 3 , 𝑏 = −1 , 𝑐 = 4, 𝑑 = 2
|3(−1) − (−2) + 4(0) + 2|
𝑑=
√(3)2 + (−1)2 + (4)2
|1|
𝑑=
√26
1
𝑑= 𝑢𝑛𝑖𝑡𝑠
√26
QUESTION 2
QUESTION 2.1
𝐿𝑒𝑡: 𝑣→ = 〈𝑎, 𝑏〉
𝑣→ ∙ 〈3, −1〉 = 0
〈𝑎, 𝑏〉 ∙ 〈3, −1〉 = 0
3𝑎 − 𝑏 = 0
𝑏 = 3𝑎
〈𝑎, 𝑏〉 = 〈𝑎, 3𝑎〉 = 𝑎〈1,3〉
〈𝑎, 𝑏〉 = 〈1,3〉
|𝑎, 𝑏| = √(1)2 + (3)2 = √10
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