# Math ia vectors

## Math Insight

Vectors are used to represent quantities that have both a magnitude and a direction. Good examples of quantities that can be represented by vectors are force and velocity. Both of these have a direction and a magnitude. A force of say 5 Newtons that is applied in a particular direction can be applied at any point in space.

In other words, the point where we apply the force does not change the force itself. Forces are independent of the point of application. To define a force all we need to know is the magnitude of the force and the direction that the force is applied in. The same idea holds more generally with vectors.

Vectors only impart magnitude and direction. This is an important idea to always remember in the study of vectors. In a graphical sense vectors are represented by directed line segments. The length of the line segment is the magnitude of the vector and the direction of the line segment is the direction of the vector. Each of the directed line segments in the sketch represents the same vector.

In each case the vector starts at a specific point then moves 2 units to the left and 5 units up. The vector denotes a magnitude and a direction of a quantity while the point denotes a location in space. Note that there is very little difference between the two dimensional and three dimensional formulas above. Because of this most of the formulas here are given only in their three dimensional version. If we need them in their two dimensional form we can easily modify the three dimensional form. There is one representation of a vector that is special in some way. So, when we talk about position vectors we are specifying the initial and final point of the vector. Position vectors are useful if we ever need to represent a point as a vector. Next, we need to discuss briefly how to generate a vector given the initial and final points of the representation. Note that we have to be very careful with direction here.EDIT: These notes were published in These notes are what should be remembered in my humble opinion going into the exam: useful facts or relationships that will save you time.

Disclaimer: mathematics is one of my passions, and I really like the subject. As such, most of this was floating around my head. I made these notes partially to remind myself, but mostly for the benefit of my peers to whom I distributed these notes. Obviously, to practice this, you ought to do past papers. But do them actively, making sure you understand what you are doing I have recently found another blog which goes through IB problems that is a great demonstration of this. Here is the link: HL Maths notes. These notes were made specially for the statistics option. However, I think it would be productive for you to read it anyway, and just be clear on which topics are core and which are option. That being said, I have made them fairly detailed and there has been a generally positive response among those who have used the notes.

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Wolfram Alpha. American Mathematical Society. Internet Suffixes. Visualead QR Generator. Desmos Calculator.A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position without rotating itthen the vector we obtain at the end of this process is the same vector we had in the beginning.

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Two examples of vectors are those that represent force and velocity. Both force and velocity are in a particular direction. The magnitude of the vector would indicate the strength of the force or the speed associated with the velocity. We won't need to use arrows here. When we want to refer to a number and stress that it is not a vector, we can call the number a scalar.

You can explore the concept of the magnitude and direction of a vector using the below applet. Note that moving the vector around doesn't change the vector, as the position of the vector doesn't affect the magnitude or the direction. But if you stretch or turn the vector by moving just its head or its tail, the magnitude or direction will change.

This applet also shows the coordinates of the vector, which you can read about in another page. The magnitude and direction of a vector. The two defining properties of a vector, magnitude and direction, are illustrated by a red bar and a green arrow, respectively. More information about applet. There is one important exception to vectors having a direction.

Since it has no length, it is not pointing in any particular direction. There is only one vector of zero length, so we can speak of the zero vector.

IB Math - Vectors - Equation of a line

We can define a number of operations on vectors geometrically without reference to any coordinate system. Here we define additionsubtractionand multiplication by a scalar. On separate pages, we discuss two different ways to multiply two vectors together: the dot product and the cross product.

Recall such translation does not change a vector. The vector addition is the way forces and velocities combine.This is a basic, though hopefully fairly comprehensive, introduction to working with vectors. Vectors manifest in a wide variety of ways from displacement, velocity, and acceleration to forces and fields.

This article is devoted to the mathematics of vectors; their application in specific situations will be addressed elsewhere. A vector quantityor vectorprovides information about not just the magnitude but also the direction of the quantity. When giving directions to a house, it isn't enough to say that it's 10 miles away, but the direction of those 10 miles must also be provided for the information to be useful.

Variables that are vectors will be indicated with a boldface variable, although it is common to see vectors denoted with small arrows above the variable. Just as we don't say the other house is miles away, the magnitude of a vector is always a positive number, or rather the absolute value of the "length" of the vector although the quantity may not be a length, it may be a velocity, acceleration, force, etc. A negative in front a vector doesn't indicate a change in the magnitude, but rather in the direction of the vector.

In the examples above, distance is the scalar quantity 10 miles but displacement is the vector quantity 10 miles to the northeast. Similarly, speed is a scalar quantity while velocity is a vector quantity. A unit vector is a vector that has a magnitude of one. The unit vector xwhen written with a carat, is generally read as "x-hat" because the carat looks kind of like a hat on the variable. The zero vectoror null vectoris a vector with a magnitude of zero. It is written as 0 in this article.

Vectors are generally oriented on a coordinate system, the most popular of which is the two-dimensional Cartesian plane. The Cartesian plane has a horizontal axis which is labeled x and a vertical axis labeled y. Some advanced applications of vectors in physics require using a three-dimensional space, in which the axes are x, y, and z. This article will deal mostly with the two-dimensional system, though the concepts can be expanded with some care to three dimensions without too much trouble.

Vectors in multiple-dimension coordinate systems can be broken up into their component vectors. In the two-dimensional case, this results in a x-component and a y-component. When breaking a vector into its components, the vector is a sum of the components:. Note that the numbers here are the magnitudes of the vectors. We know the direction of the components, but we're trying to find their magnitude, so we strip away the directional information and perform these scalar calculations to figure out the magnitude.

Further application of trigonometry can be used to find other relationships such as the tangent relating between some of these quantities, but I think that's enough for now. For many years, the only mathematics that a student learns is scalar mathematics. If you travel 5 miles north and 5 miles east, you've traveled 10 miles.You can edit the text in this area, and change where the contact form on the right submits to, by entering edit mode using the modes on the bottom right.

You can set your address, phone number, email and site description in the settings tab. Link to read me page with more information. IAs take way too long to get done, and are the easiest of things to procrastinate on.

Like all that coursework you actually have to do. The Math IA is the king of this procrastination.

### Reasonable Deviations

Utilizing the correct terminology and applying it effectively to a real-life example requires a new framework for thinking, one that has to be learnt from scratch. It should not take you hours and hours to write. You have better things to do, and if it's consuming your weekends, you're doing something wrong. It should be bringing your grade up. There is no excuse for your IA to be the reason you don't get the grade you need. It needs to be your lifeline.

However, using the strategies outlined in this book, he was able to receive a solid 7 on his Math IA.

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All you need is the right preparation, the proven tactics, and a mindset shift. His Math IA grade was the reason he got accepted into his top-choice university. Why not also make it your lifeboat? Contact Us Use the form on the right to contact us. IB Survivors. Info Email.

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Math IA. Let me ask you a few questions. If your Math IA doesn't accomplish the following, I think you might have a problem:. The Solution. Proof that It works:. Here's a sample of what you'll be getting:. Get it now Only The length of the line shows its magnitude and the arrowhead points in the direction. The two vectors the velocity caused by the propeller, and the velocity of the wind result in a slightly slower ground speed heading a little East of North.

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that. Velocityaccelerationforce and many other things are vectors. The vector a is broken up into the two vectors a x and a y.

When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is. And now you know why numbers are called "scalars", because they "scale" the vector up or down. You can read how to convert them at Polar and Cartesian Coordinatesbut here is a quick summary:. And we have this rounded result:. And it looks like this for Sam and Alex:. Hide Ads About Ads. Vectors This is a vector: A vector has magnitude size and direction : The length of the line shows its magnitude and the arrowhead points in the direction. We can add two vectors by joining them head-to-tail: And it doesn't matter which order we add them, we get the same result: Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.

If you watched the plane from the ground it would seem to be slipping sideways a little. Example: k b is actually the scalar k times the vector b. A vector can also be written as the letters of its head and tail with an arrow above it, like this:.

How do we multiply two vectors together? There is more than one way! The scalar or Dot Product the result is a scalar. The vector or Cross Product the result is a vector. Read those pages for more details. Author: Tojasho