# A few questions on Linear Algebra

I actually posted this question a few weeks ago where I wanted my solutions to a few Linear Algebra questions checked. Now thanks to useful links provided by @GerryMyerson I can ask my questions (and verify my answers).

The first three questions I just want solution verifications and suggestions for methods which are faster than the ones I used (also no answers were provided by the creator, so i'm not even sure of the correct answer).

Question 1:

Show that the equation:

$$x^2 + y^2 + z^2 + 8x -6y + 2x + 17 = 0$$

represents a sphere, and find its centre and radius.

My Solution:

$$x^2 + y^2 + z^2 + 8x -6y + 2x + 17 = 0$$

By completing the square we have that:

$$((x+4)^2 -16) + ((y-3)^2 -9) + ((z+1)^2 -1) + 17 = 0$$

Simplifying and collecting we have that:

$$(x+4)^2 + (y-3)^2 + (z+1)^2 = 9 .$$

Since a sphere is an equation of the form:

$$(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = r^2$$ where $$x_0,y_0,z_0 = (-4,3,-1)$$ are the coordinates of the centre of the sphere. Since $$r^2 = 9$$ the radius is $$3$$. Our expression can be rewritten as:

$$(x-(-4))^2 + (y-(3))^2 + (z-(-1))^2 = 0,$$ which is indeed the representation of a sphere (I think).

Question 2:

Find the area of the triangle with vertices $$P(1,-2,3), Q(0,3,1)$$ and $$R(-1,1,0)$$.

My Solution:

Since $$\vec{PQ} = \langle -1,5,-2 \rangle$$ and $$\vec{PQ} = \langle -2,3,-3 \rangle$$ we take $$\vec{PQ} \cdot \vec{QR}$$ and have that:

$$\begin{vmatrix} i & j & k\\ -1 & 5 & -2 \\ -2 & 3 & -3 \end{vmatrix}$$

Simplifying we get: $$-9i + j + 7k$$ therefore $$\vec{PQ} \cdot \vec{QR} = \langle -9, 1, 7\rangle$$.

$$\Vert \langle \vec{PQ} \cdot \vec{QR} \rangle \Vert = \sqrt{(-9)^2 + (1)^2 + (7)^2} = \sqrt{131}$$. Therefore using the formula $$A = \frac{1}{2}\Vert \vec{v} \cdot \vec{u} \Vert$$ we get:

$$A = \frac{1}{2} \cdot \sqrt{131}$$ $$A = 0.5 \cdot 11.4455$$ $$A = 5.723$$ units squared.

Question 3:

Find the volume of the paralelpiped spanned by the vectors $$a = \langle 1,2,3 \rangle$$, $$b = \langle 0,1,-1 \rangle$$ and $$c = \textbf{i} + \textbf{j}$$

My Solution:

Vector $$\textbf{c}$$ can be rewritten as $$\langle 1,1,0 \rangle$$. Taking the $$3\times 3$$ matrix we get that:

$$\begin{vmatrix} 1 & 2 & 3\\ 0 & 1 & -1 \\ 1 & 1 & 0 \end{vmatrix} = 1 \begin{vmatrix} 1 & -1 \\ 1 & 0 \end{vmatrix} - 2 \begin{vmatrix} 0 & -1 \\ 1 & 0 \end{vmatrix} + 3 \begin{vmatrix} 0 & 1 \\ 1 & 1 \end{vmatrix} = \vert -4 \vert$$

Simplifying the determinant we get that the volume is $$4$$ units cubed.

Those were the three question's that I need method/solution verification and improvements (if needed) for. I am still a beginner so if you do have any advice could you please explain it in a more fundamental way.

• Near the end of question 1 you got the signs wrong when you tried to fit each into $(v-k)^2$ form where $v$ is a variable and $k$ a constant. Nov 15, 2021 at 4:27
• @coffeemath: Should it be $(x-(-4))^2 + (y-(+3))^2 + (z-(-1))^2$ then? Nov 15, 2021 at 4:28
• In solution 2, I think you mean $\vec{QR} = \langle -2,3,-3 \rangle$...and "we take $\vec{PQ} \times \vec{QR}$" (a typo, your procedure is correct) and $\vec{PQ} \times \vec{QR}=-9\hat{i}+1\hat{j}+7\hat{k}$. Nov 15, 2021 at 4:32
• @Ajay I don't know, but would hazard guess that the issue may be the explicitly "compound" nature of the question (i.e., multiple questions are included under a single heading/post). Although people are often naturally reluctant to attempt to post multiples (e.g. because they do not want to be seen as attempting to bother people, or accumulate multiples of reputation upvotes for related queries) there is a site-wide preference for single-subject posts. Putting that issue aside your attempts generally look OK to me and definitely reflect the amount of effort that people like to see. Best wishes Nov 15, 2021 at 6:40
• @Wolgwang: Thanks! Nov 15, 2021 at 8:41