Math & Data Science Q&A - Get answers from our AI that are verified by human experts
First time here? Checkout the FAQs!

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

MathsGee Android Q&A

0 like 0 dislike
Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.
a) For which value(s) of the real scalar \(c\) is the set
E:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c=0\right\}
(an ellipsoid) non-empty?

b) For what value(s) of the scalar \(d\) is the plane \(Z:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{N}, \vec{x}\rangle=d\right\}\) tangent to the above ellipsoid \(E\) (assumed non-empty)?

[SUGGESTION: First discuss the case where \(A=I\) and \(\vec{b}=0\). Then show how by a change of variables, the general case can be reduced to this special case. ]
in Mathematics by Platinum (164,226 points) | 250 views

1 Answer

0 like 0 dislike
Best answer
(a) \(c \leq\left\langle b, A^{-1} b\right\rangle\). If \(n=1\), this of course reduces to a familiar condition.

(b) \[
d=-\left\langle\vec{N}, A^{-1} \vec{b}\right\rangle \pm \sqrt{\left\langle\vec{N}, A^{-1} \vec{N}\right\rangle} \sqrt{\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle-c} .
For \(n=1\) this is just the solution \(d=\frac{-b \pm \sqrt{b^{2}-a c}}{a}\) of the quadratic equation \(\left.a x^{2}+2 b x+c=0 .\right]\)
by Platinum (164,226 points)

Related questions

0 like 0 dislike
0 answers
0 like 0 dislike
0 answers
1 like 0 dislike
1 answer

Join MathsGee and get expert-verified answers to your maths and data science questions fast. We use a combination of generative AI and human experts to provide you the best answers to your questions. Ask a question now!

On the MathsGee, you can:

1. Ask and answer questions

2. Get expert-verified answers

3. Vote on questions and answers

4. Tip your favorite community members

5. Join expert live video sessions (Paid/Free)

6. Earn points by participating

7. Start a Fundraiser

Posting on MathsGee

1. Remember the human

2. Act like you would in real life

3. Find original source of content

4. Check for duplicates before publishing

5. Read the community guidelines

MathsGee Rules

1. Answers to questions will be posted immediately after moderation

2. Questions will be queued for posting immediately after moderation

3. Depending on the number of messages we receive, you could wait up to 24 hours for your message to appear. But be patient as posts will appear after passing our moderation.

MathsGee Android Q&A

MathsGee Android Q&A