@?public

What is $\hat{i}$?? The unit vector in the x direction. What is $\hat{j}$?? The unit vector in the y direction. What is the unit vector in the x direction?? $$ \hat{i} $$ What is the unit vector in the y direction?? $$ \hat{j} $$ What is $\begin{matrix}-2 \ 3\end{matrix}$ in unit vector form?? $$ -2\hat{i} + 3 \hat{j} $$ What is $4\hat{i} - 2\hat{j}$ in column vector form?? $$ \begin{matrix}4 \ -2\end{matrix} $$ What is the magnitude of displacement?

Resolving Vectors This links to [[Physics - Combining Vectors]]S , since it’s like doing it in reverse. What is resolving vectors?? Finding the horizontal and vertical component of a vector. What is resolving vectors opposite to…?? Combining a vector. How is the the horizontal component of $F$ written?? $F_x$. What is $F_x$?? The horizontal component of vector $F$. How is the the vertical component of $F$ written?? $F_y$. What is $F_y$?? The vertical component of vector $F$.

Combining Vectors Combining Vectors is [[Physics - Resolving Vectors]]S in reverse, or vice-versa. What is the golden rule for adding vectors?? Always add vectors tip to tail. Combine these vectors, a boat moving $10kmh^{-1}$ across a $10kmh^{-1}$ flowing river PHOTO?? PHOTO $\sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2} kmh^{-1}$ When can Pythagoras be used to combine two vectors?? When two known forces are at a right angle. How can you work out the angle between two forces?

How do you find the intersection for something like $y = 5 - 2x$ and $y = 3x^2 + 5x - 3$?? Set them equal to each other and simplify. $5 - 2x = 3x^2 + 5x - 3$ What’s the most amount of times a straight line and a quadratic could intersect?? What’s the least amoount of times a straight line and a quadratic could intersect?? A line of the form $y = mx+c$ with one point of intersection with a quadratic has what relationship?

What is a linear model?? A model which works with a straight line. When creating a linear model, how do you need to change $y = mx + c$?? Instead of using $y$ and $x$ you use variables which reflect the context. What is the general ice cream sales equation?? $\text{ice cream sales} = m \times \text{weather} + c$ What’s a common critique of a linear model?? It predicts impossible negative values. What does the gradient in a linear model reflect?

When working with triangles in a co-ordinate system, the most important step is?? Drawing a sketch. How can you prove that a triangle is right-angled in a co-ordinate system?? Find the gradients of the lines and prove two are perpindicular. How can you prove two lines are perpindicular?? Show their gradients multiply to give $-1$. How can you prove that a triangle is isoceles in a co-ordinate system?? Show two lines have the same length.

The sole purpose of this entry being seperate is so that I don’t have to re-load the other [[Further Maths - Series]]S into Anki because it’s got some formatting issues that are down to underscores inside Latex formulas being recognised as italics rather than subscripts. How can you rewrite $\sum^{n}_{r=1} (r+4)^3$?? $$ \sum^{n+4}{r=1} r^3 + \sum{4}{r=1} r^3 $$ Why can you rewrite $\sum^{n}_{r=1} (r+4)^3$?? Because it’s the same as a $r^3$ sequence starting at $5$.

Programming Languages What is a low level programming language?? A programming language that provides little or no abstraction of a computer’s instruction set. What are the two parts of an instruction in machine code?? The opcode The operand What is the purpose of having an opcode and an operand?? What do you have (operand) and what do you want do with it (opcode). What is an instruction set?? All the instructions that a computer can understand and exectue.

What is kinematics?? Analysing the motion of an object (without caring about what causes the motion). Why would $10.1 \pm 0.5ms^{-2}$ be an acceptable value for $g$?? Because the actual value of $g$ is $9.81ms^{-2}$, so the result is within the range of uncertainty. What did Galileo predict about the motion of a falling object?? That its acceleration is uniform. What is the equation for the velocity of a falling object?? $$ v = at $$

A different view of the mind There’s this really interesting passage from the Gwern essay LSD Microdosing RCT which I felt motivated enough to take brief notes on. Turns out that this is actually a snippet from a transcript of a podcast by Sam Harris, the author of Waking Up.. What does ‘The Doors of Perception’ argue the brain’s primary function is?? Eliminative, filtering out a potential vast, transpersonal dimension of mind. Why could an eliminative brain be neccessary?

What is the formula for the sum of the cubes of the first $n$ natural numbers?? $$ \sum^{n}_{r=1} r^3 = \frac{1}{4}n^2(n+1)^2 $$ How could you rewrite $\sum^{n}_{r=1} r^3$?? $$ \frac{1}{4}n^2(n+1)^2 $$ What’s another way of expressing $\frac{1}{4}n^2(n+1)^2$?? $$ \sum^{n}_{r=1} r^3 $$ How could you rewrite $\sum^{n}_{r=1} 4r^2$?? $$ n^2(n+1)^2 $$ What’s an easy way for remembering the sum of cubes formula?? It’s the sum of the natural numbers formula squared. Backlinks [[Further Maths - Series]]S [[Further Maths - Syllabus]]S Metadata date: 2020-09-16 17:57 tags: - '@?

What is the formula for the sum of the squares of the $n$ natural numbers?? $$ \sum^{n}_{r=1} r^2 = \frac{1}{6}n(n+1)(2n+1) $$ How could you rewrite $\sum^{n}_{r=1} r^2$?? $$ \frac{1}{6}n(n+1)(2n+1) $$ What’s another way of expressing $\frac{1}{6}n(n+1)(2n+1)$?? $$ \sum^{n}_{r=1} r^2 $$ How could you rewrite $\sum^{n}_{r=1} 3r^2$?? $$ \frac{1}{3}n(n+1)(2n+1) $$ Backlinks [[Further Maths - Series]]S [[Further Maths - Syllabus]]S Metadata date: 2020-09-16 17:57 tags: - '@?further-maths' - '@?series' - '@?