Exam (elaborations)
MAT1503 Assignment 5 (COMPLETE ANSWERS) 2024 - DUE 10 September 2024
- Course
- Institution
- Book
MAT1503 Assignment 5 (COMPLETE ANSWERS) 2024 - DUE 10 September 2024
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
MAT1503 Assignment 5 (COMPLETE ANSWERS) 2024 - DUE 10 September 2024 ; 100% TRUSTED Complete, trusted solutions and explanations. Ensure your success with us ....
-
MAT1503 Assignment 5 (COMPLETE ANSWERS) 2024 - DUE 10 September 2024 ; 100% TRUSTED Complete, trusted solutions and explanations. Ensure your success with us ....
-
MAT1503 Assignment 5 (COMPLETE ANSWERS) 2024 - DUE 10 September 2024 ; 100% TRUSTED Complete, trusted solutions and explanations. Ensure your success with us ....
Seller
Follow
THEBLAZE1
Reviews received
Content preview
,MAT1503 Assignment 5 (COMPLETE ANSWERS)2024 - DUE 10 September 2024 ; 100% TRUSTED
Complete, trusted solutions and explanations.
Question 1: 12 Marks (1.1) Let U and V be the planes given by:
(2) U : λx + 5y − 2λz − 3 = 0, V : −λx + y + 2z + 1 = 0. Determine
for which value(s) of λ the planes U and V are: (a) orthogonal,
(2) (b) Parallel. (2) (1.2) Find an equation for the plane that
passes through the origin (0, 0, 0) and is parallel to the (3) plane
−x + 3y − 2z = 6. (1.3) Find the distance between the point
(−1,−2, 0) and the plane 3x − y + 4z = −2. (3)
1.1 Planes U and V
Planes:
U:λx+5y−2λz−3=0U: \lambda x + 5y - 2\lambda z - 3 =
0U:λx+5y−2λz−3=0
V:−λx+y+2z+1=0V: -\lambda x + y + 2z + 1 = 0V:
−λx+y+2z+1=0
a) Orthogonal Planes
To determine when two planes are orthogonal, we need to
check when their normal vectors are orthogonal.
Normal Vector of Plane U: nU=(λ,5,−2λ)\mathbf{n_U} = (\
lambda, 5, -2\lambda)nU=(λ,5,−2λ)
Normal Vector of Plane V: nV=(−λ,1,2)\mathbf{n_V} = (-\
lambda, 1, 2)nV=(−λ,1,2)
,Two vectors are orthogonal if their dot product is zero. Thus, we
compute the dot product of nU\mathbf{n_U}nU and nV\
mathbf{n_V}nV:
nU⋅nV=(λ)(−λ)+(5)(1)+(−2λ)(2)\mathbf{n_U} \cdot \mathbf{n_V}
= (\lambda)(-\lambda) + (5)(1) + (-2\lambda)(2)nU⋅nV=(λ)(−λ)
+(5)(1)+(−2λ)(2) nU⋅nV=−λ2+5−4λ\mathbf{n_U} \cdot \
mathbf{n_V} = -\lambda^2 + 5 - 4\lambdanU⋅nV=−λ2+5−4λ
Set the dot product to zero for orthogonality:
−λ2+5−4λ=0-\lambda^2 + 5 - 4\lambda = 0−λ2+5−4λ=0
Rearrange:
λ2+4λ−5=0\lambda^2 + 4\lambda - 5 = 0λ2+4λ−5=0
Solve this quadratic equation using the quadratic formula:
λ=−b±b2−4ac2a\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}
{2a}λ=2a−b±b2−4ac
Here a=1a = 1a=1, b=4b = 4b=4, and c=−5c = -5c=−5:
λ=−4±16+202\lambda = \frac{-4 \pm \sqrt{16 + 20}}
{2}λ=2−4±16+20 λ=−4±362\lambda = \frac{-4 \pm \sqrt{36}}
{2}λ=2−4±36 λ=−4±62\lambda = \frac{-4 \pm 6}{2}λ=2−4±6
Thus:
λ=22=1orλ=−102=−5\lambda = \frac{2}{2} = 1 \quad \text{or} \
quad \lambda = \frac{-10}{2} = -5λ=22=1orλ=2−10=−5
b) Parallel Planes
, For the planes to be parallel, their normal vectors must be
scalar multiples of each other:
(λ,5,−2λ) and (−λ,1,2)(\lambda, 5, -2\lambda) \text{ and } (-\
lambda, 1, 2)(λ,5,−2λ) and (−λ,1,2)
We need to find if there exists a scalar kkk such that:
(λ,5,−2λ)=k(−λ,1,2)(\lambda, 5, -2\lambda) = k(-\lambda, 1, 2)
(λ,5,−2λ)=k(−λ,1,2)
Equating components, we get:
λ=−kλ\lambda = -k\lambdaλ=−kλ 5=k5 = k5=k −2λ=2k-2\lambda
= 2k−2λ=2k
From 5=k5 = k5=k, substitute kkk into −2λ=2k-2\lambda =
2k−2λ=2k:
−2λ=2⋅5-2\lambda = 2 \cdot 5−2λ=2⋅5 −2λ=10-2\lambda =
10−2λ=10 λ=−5\lambda = -5λ=−5
Substitute λ=−5\lambda = -5λ=−5 into λ=−kλ\lambda = -k\
lambdaλ=−kλ:
−5=−k(−5)-5 = -k(-5)−5=−k(−5) −5=5k-5 = 5k−5=5k k=−1k = -
1k=−1
The value λ=−5\lambda = -5λ=−5 satisfies the parallel condition
with k=−1k = -1k=−1.
1.2 Equation for Plane Parallel to Given Plane
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 THEBLAZE1. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $2.50. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
50064 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now