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$\noindent This is essentially Pythagoras' theorem for $\mathbb{R}^{n}$. The result also holds in general inner product spaces with a given inner product. I'll just give the proof for $\mathbb{R}^{n}$ here though as I'm not sure if you've learnt inner products. I'll provide a link to a proof...

$\noindent I'm assuming $\vec{y}$ is a fixed vector in $\mathbb{R}^{n}$ and that $A$ is a real matrix. (The given result can also be generalised to a complex setting, or any inner product space.) Let $\vec{x}_0$ be a vector in $\sigma$ such that $\vec{y}-A\vec{x}_{0}$ is orthogonal to $\sigma$...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Since $y=x^{x}$, taking logs of both sides yields $\ln y = x\ln x$. Differentiating both sides w.r.t. $x$ yields $\frac{y^\prime}{y}=\ln x + 1$. Hence $y^\prime = y \left(\ln x + 1\right) = x^x \left(\ln x +...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread If you know your early factorials, it's easy to recognise that 5! = 120. You'll probably recognise early factorials if you do lots of perms and combs Q's. In the above method, the only inspection comes about when at the stage...

Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread Inspection. $\noindent We want $\frac{8!}{\left(8-r\right)!}=336$. So $\left(8-r\right)! = \frac{8!}{336}=120$. So $8-r = 5$, so $r=3.$

In R3, it can quickly be done by comparison of normals. In higher dimensions (also applies to R3), the planes will be parallel iff the span of the direction vectors of plane 1 is equal to the span of the direction vectors of plane 2. (Note that in higher-than-3 dimensions, planes don't have a...

If you're in Cartesian form to start with, your first method is good. If you're in point-normal form to start with, we can convert it to Cartesian form and then use your first method. The second method won't work unless the vector you use to cross with the normal is one that lies in the plane...

$\noindent If you can show that $\det \left(A\right) = \vec{a} \cdot \left(\vec{b}\times \vec{c}\right)$ (dot and cross products here), then you can use the fact that $\left \| \vec{b}\times \vec{c}\right \|$ is the area of the base parallelogram, and $\vec{v}\cdot \vec{w} = \left \|\vec{v}...

$\noindent Hint: Draw a picture. First consider it as $\lambda_2$ is held fixed in $[0,1]$ (say even $\lambda_2 = 0$) and $\lambda_1$ varies from 0 to 1: what set of points is traced out? Then consider the collection of all these points for each $\lambda_2$ in $[0,1]$. What is this collection?$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Once you've found the $T$ (it's 4 you say), we just need to find the corresponding $\theta$ (recall that $T$ and $\theta$ are \emph{not} independent; the time to hit the bottom clearly depends on $\theta$, or if we...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Differentiate this with respect to $T$: $400T^2 -\left(5T^2 -40\right)^2$ (since that's $\left(x(T)\right)^2$).$

And he doesn't do 4U, right?

A method for finding the points on two skew lines that are nearest each other is described here: https://en.wikipedia.org/wiki/Skew_lines#Nearest_Points .

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent At time $T$, we have $y(T) = -40$, since the stone is at the lake, which is 40 m below the cliff. So from the equation for $y(t)$, we find $-5T^2+20T \sin \theta=-40\Rightarrow 20T\sin \theta = 5T^2 -40$. So $400T^2...

Re: University Statistics Discussion Marathon I guess that's up to you. If you'd prefer having your own questions in specific threads, you could make separate threads for them. Depends on whether you'd prefer that or prefer having them in one thread that also includes Q's from other people.

Re: University Statistics Discussion Marathon Remember that variance is the square of standard deviation. They gave us the standard deviations of the male and female masses, so we need to square them to get their variances to use in your line about variance. Then to get the standard...

For a), the pdf needs to integrate to 1. So we have $$\begin{align*}\int_1 ^3 kz \text{ d}z &= 1 \\ \Longleftrightarrow k \left[\frac{z^2}{2}\right]_1 ^3 &= 1 \\ \Longleftrightarrow k \left(\frac{3^2}{2}-\frac{1^2}{2}\right) &= 1 \\ \Longleftrightarrow 4k &= 1 \\ \Longleftrightarrow k &=...

Re: First Year Uni Calculus Marathon

Re: First Year Uni Calculus Marathon Where on BOS do you think it got asked before? As in in some marathon from the past?

With non-integer powers, we actually end up with multi-valued things, which require specification of a Principal Value. But it is the case that one of the values out of these multiple values of a rational power will be what is specified by what De Moivre's Theorem would say. E.g. something like...