Get Your Answer Fast Within An Hour
First time here? Checkout the FAQs!
x

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

0 like 0 dislike
185 views
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 (129,882 points) | 185 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 (129,882 points)

Related questions

0 like 0 dislike
0 answers
0 like 0 dislike
0 answers

Join the MathsGee Study Questions & Answers Club where you get study and financial support for success from our community. SEARCH - ASK - LEARN


On the MathsGee Study Questions & Answers, you can:


1. Ask questions


2. Answer questions


3. Vote on Questions and Answers


4. Tip your favourite community member(s)


5. Create Live Video Tutorials (Paid/Free)


6. Join Live Video Tutorials (Paid/Free)


7. Earn points for participating



Posting on the MathsGee Study Questions & Answers


1. Remember the human


2. Behave like you would in real life


3. Look for the original source of content


4. Search for duplicates before posting


5. Read the community's rules




Q&A RULES


1. Answers to questions will be posted immediately after moderation


2. Questions will be queued for posting immediately after moderation


3. Depending on how many posts we receive, you could be waiting up to 24 hours for your post to appear. But, please be patient as posts will appear after they pass our moderation.


MathsGee Android Q&A