Dimensions are the minimum measurements required to definea point. Minimization means dimensions should be independent of another, thereby maximizing efficiency of measures. Essentially, they are planes at right angles with another. Einstein proposed the fourth dimension, which is time. The SI unit of time is â€œsecondsâ€.

â€œWhen model agencies collect height and waist, these dimensions are independent, as height cannot be defined with a waist measurement, or vice versa,â€ Jamie started, â€œThis is because height is measured vertically, and waist is measured horizontally. Time is similarly independent.â€

â€œFor example, we can say Selena Gomez is 1.65m, or that the Wizards Movie runs for 98 minutes,â€ Mandy continued.

â€œYou know how in 1 Cor 2:14, it talks of those without the Spirit interpreting things of God as foolish? Itâ€™s like the Spirit is its own dimension,â€ Mandy continued, â€œimagine a person from a 3D world sticking his fingers into a whiteboard where 2D people live. What the 2D people see would just not make sense.â€

â€œThe Holy Spirit does so much,â€ Mandy commented, â€œHe formed the universe (Genesis 1:2-3), provides revelation (1 Corinthians 2:9-13), inspired scripture (2 Peter 1:20-21), regenerates (John 3:5-7), baptizes (1 Corinthians 12:13), indwells (Romans 8:9), assures salvation (Romans 8:14-16), enables Spiritual living (Galatians 5:16-25), bestows spiritual gifts to build up the church (1 Corinthians 12:4-11), and fruits (Galatians 5:22-23).

YouTube video activity

Frequently asked questions

When you talk about minimization, what are you minimizing? Measurements.

Oh, so that's why they're at [mathjax]90^{\circ}[/mathjax] to another? Yes, because neither can be used to define the other. Think about it, no changes in the x-axis mean anything in terms of the y-axis, and vice versa.

What's a plane? A plane is a flat 2D surface. Think of it like a piece of paper.

Formative learning activity

Maps to RK1.1

Identify 3 examples of dimensions, and explain why each are a dimension.

2 Vectors, components

YouTube video activity

Scalars are measurements with magnitude/power [but no direction]. Vectors have both magnitude and direction. Magnitude refers to the size property of an object. An example of a scalar quantity is time [which doesnâ€™t have direction], and of vector is force [which has direction].

â€œSo a scalar could be a mass 50kg or energy 100J,â€ Mandy said, â€œIâ€™m not saying â€˜50kg downâ€™ or â€˜100J eastâ€™.â€

â€œIn contrast, a vector could be force 500N down, or momentum 550sN up,â€ Mandy continued.

â€œSo One Direction is a vector?â€ Blaire giggled.

â€œSo long as the one direction youâ€™re looking in is towards Christ ,â€ Mandy winked.

YouTube video activity

Frequently asked questions

What is "magnitude"? Strength, power, size: these are all appropriate synonyms used to describe magnitude.

Remember Diglet from Pokemon? One of his moves was "magnitude". On the game console, the power of this move would be randomly generated, as a number between 1-10; the higher the number, the more powerful the attack, and the more damage it does on the opponent. Same idea here.

Both scalars and vectors have magnitude. This is self evident, because without magnitude... there is... nothing [to talk about]!

I see, so this is clearly different from "direction", which is not "power" (as in magnitude), but as in... Direction!

Only vectors have direction. Scalars don't.

What exactly is force and momentum? We'll get to that soon You may have heard about these principles in simple English, but they are principles which prerequisite the knowledge about... vectors!

Learning activity

Define the difference between a scalar and vector.

Identify 3 examples of scalars, and 3 examples of vectors. Provide explanation for why each of the 6 are a scalar or vector respectively.

Vectors can be shown graphically as an arrow depicting the direction, connecting an initial point ([mathjax]A[/mathjax]) with a terminating point ([mathjax]B[/mathjax]), denoted by ([mathjax]\vec{AB}[/mathjax]), with an arrow going from [mathjax]A[/mathjax] to [mathjax]B[/mathjax] from the top. The length of the [graphical] arrow depicts its magnitude. The tip of where the arrow is pointing to is called the head, and the opposing [starting] point is the tail.

The starting side of the arrow is called the... Tail

The ending side of the arrow, where the arrow points to, is called the... Head

Learning activity

Robin Hood shoots an arrow at [mathjax]45^{\circ}[/mathjax] elevation above the ground, towards the RHS of the page. Draw this.

Vector resolution

A vector can be resolved into its components, which are lines at [mathjax]90^{\circ}[/mathjax] to another, of which the sum is equal to the original vector. For instance, the red vector can be resolved into its x-component and y-component:

What are components? Parts, constituents, making-up-the-larger-thing: these are all appropriate synonyms of the word "component".

What is resolution? Where the larger-thing, is expressed instead, in its parts/constituents/etc.

So in court, a resolution is where the parties reach a pre-court... Different type of resolution we're talking about here Same spelling, different word.

A better analogy for resolution, is a computer screen's resolution. The higher the resolution, the greater the number of parts that make it up, and therefore the more smooth it looks. A very high resolution display is the iPhone's retina display, where you cannot even make out the individual pixels.

So in the analogy, the pixel is the equivalent of the... Component

So another example is the components you find inside a computer. It's what makes it up [thereby determining its processing speed, read/write speed, graphics ability, and so forth].

Usually, this is defined as a graph with an x- and y-axis, expressed in the format [mathjax][a,b][/mathjax] where [mathjax]a[/mathjax] is the x-axis measurement, and [mathjax]b[/mathjax] is the y-axis measurement. Because vectors have direction, they need to be described with two numbers, whereas scalar quantities can be described with one.

Jamie met his unlikely clinger girlfriend, Sophie Graham MD, the resident blonde klutz of the princess clique, at the Baptist church.

â€œHey!â€ Sophie asked, â€œDo you know where QE Hospital is?â€

â€œNot sure,â€ Jamie said grabbing his z-Phone to check.

Earlier, Jamie was sitting several pews behind Sophie, and had thrown a pencil in Sophieâ€™s direction to try to grab her attention, and well, thought that perhaps this was her way of reciprocating her feelings. Little did Jamie know that Sophie knew nothing of the pencil-throwing escapade, but genuinely and independently, had feelings for him.

â€œSay if that pencil was thrown 60kmph at an 30Â° elevation, what are its components?â€ Mandy asked.

â€œOh! You can also check this is correct with the Pythagorean Theorem, that [mathjax]\sqrt((30)^2+(52)^2)=60kmph[/mathjax]: QED,â€ Mandy remarked, â€œQED, Latin for quod erat demonstrandum, meaning Iâ€™m awesome aka demonstrated what I wanted.â€

Frequently asked questions

How does the x- and y-axis relate to the concept of components? Any point in 2D, can be referred to as an x- and y-axis, in the form of [x,y], in relation to its origin (0,0).

So how do you determine this x- and y- number? Usually, you're provided with a magnitude, and an angle? You can use a combination of either the Pythagorean theorem and/or trigonometry.

What's the Pythagorean theorem? It states that in a right angled triangle, the square of its two sides, is equal to the square of the hypotenuse.

What is the hypothenuse? The longest side of a right angled triangle. It's also the side opposite the right angle.

Okay, so what's trigonometry? It is a formula that can be used, that doesn't require a right angled triangle. So it is more versatile than the Pythagorean theorem.

The words "opposite" and "adjacent", what agnles are these exactly "opposite" and "adjacent" to? In pythagorus, they refer to sides, and the way they are assigned make no difference.

In trigonometry, theta is an unknown angle, which can be arbitrarily assigned. So this is the theta that is present in the equation.

Formative learning activity

Maps to RK1.2

What are vectors? What about components?

3 Vector addition

YouTube video activity

The distinction between scalar and vector quantities is important, as vector addition cannot be achieved by simply adding the magnitude numbers together of the vector. Note also, that you cannot add vectors with scalars.

Vector addition is where two vectors are added. Vectors cannot be added by simply adding [magnitude] numbers together.

â€œIsnâ€™t adding, just adding?â€ Mandy asked, puzzled.

â€œThatâ€™s right Mandy, thatâ€™s distance. But how far have you effectively travelled though, from start to finish?â€

â€œ0m, because Iâ€™m back where I started!â€ Mandy said.

â€œRight, and thatâ€™s displacement: your net movement.â€

â€œYouâ€™re back to where you started, so thatâ€™sâ€¦ 0m.â€

â€œMy goodness ,â€ Jamie laughed, â€œThe world isnâ€™t round, the moon is made of cheese, and 1+1=0 !!â€

Vector addition can be achieved either graphically, or by addition of components. The graphical method involves attaching the vectors tail-to-head with tail-to-head, with the resultant vector going from the very tail to the very head. For instance, if we add 4N east with 3N south, graphically:

It makes sense that the resultant vector (in red) has a magnitude of 5N, because created, is a right angled triangle, and as a common right angled triangle, we know sides are 3, 4, and 5, which is the solution.

This can be quite complicated, especially when dealing with 3D vectors. The component method involves resolving a vector into its components, and then adding its components.

â€œThen, writing a vector in the form of [x-component, y-component, z-component], to add [1,2,3] and [5,5,5], this is [1+5,2+5,3+5]=[6,7,8], not requiring complex geometry,â€ Mandy commented.

Why do you say scalars can be simple addition? Scalars don't have direction, so they are in the same direction. So you can just add them.

So you can't just simply add vectors because... Just imagine if they were in the opposite direction. It'd be like adding +5m with -5m. You'd end back where you started.

Is that the same reason why you can't add vectors with scalars? Sort of, the main reason is that you can't add bananas and apples. Vectors have direction, and scalars don't.

So how do you add vectors? Two options. Graphically, or mathematically.

How do you add graphically? Attach tail-to-head with tail-to-head. And the resulting vector is from the very-tail to the very-head.

How about mathematically? Resolve both vectors into their components, and add their components.

Why would you do it mathematically, when you could just do it graphically? It's easier in the 2nd dimension, but just imagine the 3rd, and even higher dimensions.

Does it matter which vector you add first? No.

Vector subtraction involves adding the converse/negative of the subtracted vector, or alternatively, [mathjax]\vec{A}-\vec{B}=(\vec{A})+(\vec{-B})[/mathjax]. For example, if we subtract 4N east with 3N south, graphically:

What is vector subtraction? Like standard subtraction, it's the opposite of addition.

So how's that exactly done? You add the subtracted vector. So you flip the subtracted vector around the opposite direction, and add that.

The most straightforward type of vector multiplication is scalar multiplication, where a vector is multiplied by a scalar. This doesnâ€™t alter the direction, but rather, only the magnitude. As such, the magnitude is multiplied directly in using the component method, or the vector is repeated by the magnitude number of times graphically. For example, if the pink vector is multiplied by 3, the resulting blue vector is in the same direction:

Can you multiply a vector with a scalar? Yes, that's called scalar multiplication. It's where a vector is multiplied in that original direction, by the "scalar" number of times.

Can you multiply a vector with another vector? Not really. But we have created 2 artificial means, including dot product, and cross product.

What is dot product? It is a calculation that produces a scalar, namely, [mathjax]ab.cos(\theta)[/mathjax]. Essentiallly, it seeks the projection of one vector onto another. In other words, it considers only the portion of the vectors which are parallel to another.

What is cross product? Vector product? Cross product and vector product is the same thing. It is a calculation that produces a vector, namely, [mathjax]ab.sin(\theta)[/mathjax]. Essentially, it is the opposite of dot product, considering only the portion of the vectors which are perpendicular to another.

So you said the cross product produces a vector. Has it got a direction? Yeah it does! This is determined by the right-hand rule, such that the index and middle fingers [of the right hand] is placed in the direction of the crossing vectors, and the resultant vector is in the directon of the thumb.

So as you can see, the cross product is perfectly perpendicular to all crossing vectors.

How can vector multiplication identify parallel and perpendicular vectors? When cross product is zero, the vectors are parallel. That's because cross product considers only the perpendicular portion, so when 0, there must be no perpendicular portion.

When dot product is zero, the vectors are perpendicular. That's because dot product considers only the parallel portion, so when 0, there must be no parallel portion.

Formative learning activity

Maps to RK1.3

What is vector addition?

4 Speed, velocity

YouTube video activity

Displacement is the vector version of distance [which is scalar]. Speed is defined as [mathjax]\dfrac{distance}{time}[/mathjax], and is hence scalar. Velocity is defined as [mathjax]\dfrac{displacement}{time}[/mathjax], and is hence the vector version of speed. Displacement is the shortest/effective distance from the initial to final position, whereas distance is the actual path taken. The SI unit of distance is â€œmetersâ€.

â€œIf you fly from London to NYC, the plane doesnâ€™t fly in a straight line! It twists and turns,â€ Jamie said.

â€œSo even though the displacement from London to NYC is 5585km,â€ Mandy said, â€œIâ€™m going to be travelling a greater than this, because of the twists. The total distance Iâ€™ve travelled is referred to as distance.â€

â€œThe effect is that though speed may be a non-zero number, because velocity uses displacement, if we travel from London to NY and back, velocity will be 0m/s.â€

Note that velocity is a property that doesnâ€™t depend on quantity, known as an intensive property. In contrast, a property that does depend on quantity is known as an extensive property, for example, mass and energy.

Frequently asked questions

What's the difference between displacement and distance? Displacement is a vector, distance is a scalar. Displacement is the shortest distance from 2 points, whereas distance is the actual path taken.

What's the difference between velocity and speed? Velocity is a vector, speed is a scalar. Velocity uses displacement, and speed uses distance. Both have the same structure [mathjax]s=\dfrac{d}{t}[/mathjax], however.

What's the difference between an intensive and extensive property? Intensive properties don't depend on quantity, whereas extensive properties do [depend on quantity]. Speed for example, doesn't increase simply because there is more of that object. Mass however, does.

Formative learning activity

Maps to RK1.4

What is speed? What is velocity, and how does is it distinct from speed?

What is deceleration? It refers to acceleration that is negative. That is, a reduction in velocity.

Formative learning activity

Maps to RK1.5

What is acceleration?

6 Free falling bodies

YouTube video activity

Free falling is the permission of an object to be subject to gravitational acceleration. Gravity is an acceleration downwards of [mathjax]10m/s^2[/mathjax] (in physics, the figure more generally used is [mathjax]9.8m/s^2[/mathjax]).

â€œNewton founded gravity, because one of â€˜em apples dropped on his head, and he wondered: why?â€ Mandy said.

â€œHow â€˜bout them apples,â€ Blaire giggled.

â€œI think I now know why I like red apples,â€ Mandy said, â€œthem apples got product placement in Snow White .â€

Frequently asked questions

What is free falling? Permitting an object to be subjected only to gravity.

Where the object is not dropped, for example, thrown up at [mathjax]50m/s[/mathjax], gravity will still act, but in the opposite direction (at least at the start). The time taken for the object to reach its maximum height is 5 seconds, as the object slows [mathjax]10m/s[/mathjax] each second. The time taken for it to reach its maximum, back to its original height, is that time multiplied by two. So for an object thrown up at [mathjax]50m/s[/mathjax], it will take 5 seconds up, and 5 seconds down, so 10 seconds in total.

â€œI canâ€™t seem to find it,â€ Jamie replied; realizing his stoush at Sophie was, well, free falling.

â€œBy the way,â€ Jamie said, â€œmy name is Jamie.â€

â€œMy real name is Sophia, but people call me Sophie.â€

â€œWhatâ€™s with the massive Disney smile ?â€ Jamie asked.

â€œDo I look like the girl from Tangled?â€ Sophie giggled.

Realizing that he couldnâ€™t just continue fidgeting his phone, pretending to find a non-existent hospital, he gathered himself, preparing his next statement.

â€œHey, did you get that pencil?â€ Jamie asked.

â€œWhat pencil?â€ Sophie remarked, confused.

And that was it. Jamie realized it was just all in his head; she was not interested in him, and just genuinely wanted to find this hospital.

â€œI canâ€™t find it,â€ Jamie sighed, trying to figure if there was something he could ask her, just so this magical moment could be prolonged. But he had nothing. Or at least nothing came out.

â€œAlright,â€ Sophie replied after a moment of silence, â€œthanks for that Jamie, Iâ€™ll see you around!â€

Secretly, like Jamie, Sophie also wanted their conversation to prolong, but she didnâ€™t really know what to say. After all, she was the princess clique type girl, and everybody knows that they donâ€™t say much.

Frequently asked questions

What is instantaneous velocity? Velocity at any particular point in time. For example, the velocity after a certain amount of time, after it is thrown upwards, permitted to free fall, or thrown as a projectile.

What is average velocity? Just as it sounds. The mean velocity.

What are the uniform acceleration motion equations? Equations that can be used when acceleration is uniform (i.e. constant). They are a series of formulas, that link displacement, initial and final velocity, acceleration, and time.

Projectile motion is where there is not only a vertical component, but also a horizontal component present. Keep in mind that the components are independent of each other, and gravitational acceleration only acts on the vertical component, as it is a force downwards. As such, the motion can be separated into a vertical and horizontal component, and equations applied on to these components independent of each other. For example, acceleration of the vertical component is [mathjax]10m/s^2[/mathjax] down, but acceleration of the horizontal component is [mathjax]0m/s^2[/mathjax]. The connecting factor is time. For example, a projectile shot at [mathjax]100m/s[/mathjax] at an [mathjax]30^{\circ}[/mathjax] angle above ground.

How does projectile motion differ from free falling? Free falling was just a drop (i.e. vertical movement). Projectile motion is where there's also a horizontal component present.

How do we answer projectile motion questions? Remember that the vertical and horizontal components are independent.

However, there are some things that are shared. For example, the time is shared. The vertical and horizontal components don't have different times. When the object starts, it starts. When it stops, it stops.

What is range? The horizontal distance travelled.

Although mass and shape do not alter velocity per se, they do affect air resistance, which in turn alters velocity. Air resistance is a type of drag. Drag is a force which acts on an object in the opposite direction of velocity, due to air molecules. For the example of air resistance, the greater amount of air molecules hit, the greater the air resistance. Thus, shape (determining surface area) and velocity affect air resistance. Mass doesnâ€™t affect the force of air resistance, but does affect the ability of air resistance to act against the momentum of the object.

Frequently asked questions

But don't mass and shape affect velocity? Not directly. They affect air resistance, which in turn affect velocity.

What is air resistance? It's a type of drag. Drag is a force caused due to air molecules, opposing the direction of velocity.

So what's exactly the effect of shape on air resistance? Increased shape increases air resistance.

How about mass - how does that affect air resistance? Mass doesn't affect the force of air resistance. However, it does affect the ability of air resistance to oppose the momentum of the object.

Formative learning activity

Maps to RK1.6

What is a free falling body?

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