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$In two dimensions, let $\textbf{a}=\left[\begin{matrix}a_1\\a_2\end{matrix}\right]$ and $\textbf{b}=\left[\begin{matrix}b_1\\b_2\end{matrix}\right]$ and consider the parallelogram spanned by $\textbf{a}$ and $\textbf{b} \textbf{Show}$ that a parametric vector form for the parallelogram is...

Question is easy but I cannot find my mistake. $Find the shortest distance between the line through the points $(1,3,1), \, (1,5,-1)$ and the line through the points $(0,2,1), \, (1,2,-3) Not sure if this is the standard method but this is according to the method I was taught. \lambda, \mu...

Are you happy?

$Suppose I have these vectors:$\\ \overrightarrow { AB } =\left[ \begin{matrix} 2 \\ 2 \\ -1 \end{matrix} \right] \\ \overrightarrow { BC } =\left[ \begin{matrix} -1 \\ -1 \\ 5 \end{matrix} \right] \\ Ignore CA for now $I'm required to find the cosine of the internal angles of $\triangle...

Starting with a dumb question because I think my lecturer forgot to explicitly say how to. I know what the graph looks like already r=6\sin{\theta} But how do I convert this to Cartesian form lol

I've forgotten why L'Hopitals breaks down for this. \lim_{x\rightarrow\infty}{\frac{x+\sin{x}}{x-\sin{x}}} Isn't it technically infty/infty

The MATLAB software I downloaded at home supposedly has a licence expiry date set within 12 days from now. How do I resolve this issue? (If there's already a post on this please link) Thanks

Already known: \det{\left( \begin{matrix} z & 1 & 2 \\ 1 & z & 3 \\ 1 & 1 & z+1 \end{matrix} \right) }=\left(z-1\right)\left({z}^{2}+2z-4\right) And hence solve: zx+y=2 x+zy=3 x+y=z+1 Can't draw the link between the statements. All I get is this: \text{By using the exact same...

Let U1 and U2 be two nxn row-echelon matrices. Prove that det(U1)det(U2)=det(U1U2) So far... \\$Let the components of ${U}_{1}, U_{2}, U_{1}U_{2}$ respectively be ${u}_{ij},{v}_{ij},{a}_{ij}\\$ for some integers $i, j \in [1,n] \\$Since ${U}_{1}$ and ${U}_{2}$ are upper triangular:$\\\left| {...

So the question just said to show that Q = (cos, -sin // sin cos) is orthogonal, and that x (in R2) and Qx are equidistant. Easy. What does it mean here to say that Q acts as a rotation on R2?

$Let $f$ be continuous on $[a,b]$. Prove that there exists $c\in\(a,b)$ such that$\\ \int_{a}^{b}{f(t)dt}=f(c)(b-a) When I tried experimenting with the MVT I found something like this: f(b)-f(a)=f^{\prime}(c)(b-a) However they gave a hint and I have no idea how to apply it. $Hint: Let...

If f' is (continuous, differentiable and) monotone, f always concaves the same way

So I got part a) out. Need only a small starter on part b). But it's mainly cause of the process. Q: Use the MVT to prove that a) ln(1+x)<x for x>0 b) -ln(1-x)<x/(1-x) for 0<x<1 So for part a) I defined f:[0,x], f(t)=ln(1+t) and used the MVT to get ln(1+x)/x=1/c (for some c in (0,x)) So cause...

These are just when at least one left or right side limit is undefined right? So obviously sin(1/x) has this problem at x=0 due to the rapidly oscillating behaviour. Would vertical asymptotes also get classified as this?

This is preposterous... What's the easiest way to do this question? $If $A$ and $B$ are square matrices such that $AB=BA$ show that $\\(A-B)(A+B)={A}^{2}-{B}^{2} Writing out the components takes way too long

Posting this in the 2U section as it's still within the scope of the course (even though it's just my homework). $The function $f$ is defined by $f(x)={x}^{2}-4x+3$ for $0 < x \le 3$, $f(-x)=-f(x)$ for all $x$ and $f(x+6)=f(x)$ for all $x$. I have the graph in front of me so I can pretty much...

LaTeX is broken again so I won't use it Am I doing this correctly? Q: As h->0, show that (explain why) sin(x+h)=sinx+hcosx+o(h) A: Let g(x)=sin(x+h)-sinx-hcosx Need |g(x)|<=K|h| for all K>0 i.e. |sin(x+h)-sinx-hcosx|<=K|h| <=> | (sin(x+h)-sinx)/h - cosx| <= K (as h neq 0) LHS = |...

Can't start this question cause I don't know what I'm looking at. How would I simulate it? $In this problem we shall calculate the area of a spherical triangle. Consider the surface of a sphere of unit radius of area $4\pi$. A great circle on a sphere is the intersection of that sphere with a...

Just start me off please. No need to complete the question. $Assume 3 countries $A, B, $ and $C$ trade with one another and no-one else, that a common currency is used and that each country's total income comes from trade with the others or sales to itself and nothing else.$ \\ A$ spends...

In other words, could you be directly asked on one of the following in 2U (and I guess any other relevant HSC courses)? \\\frac{d}{dx}\int_{c}^{x}{f(t)\,dt}=f(x) $+ a chain rule application:$\\\frac{d}{dx}\int_{c}^{g(x)}{f(t)\,dt}=f'(g(x))g'(x)