Exam (elaborations)
MAT1503 Assignment 8 with Complete Solutions 2023
- Course
- Institution
MAT1503 Assignment 8 with Complete Solutions 2023
[Show more]
Seller
Follow
BESTGRADED
Reviews received
Content preview
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller BESTGRADED. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.00. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
66475 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now