math
8th Grade - Vectors

8th Grade lesson

10 - Vectors, maps


Vectors

A vector, it's this : vector
It is an arrow. You can draw a vector if you know its length, its direction and its end (the end of the arrow). You can give it a name, for example call vector u the vector above.
It is not useful to know where it comes from. They are all equal. The image B of a point A by the translation of vector translation, is the point which is at the end of the vector when the origin of the vector is point A.
translation

A vector is kind of like an arrow indicating a movement. If you go up and on the right, at the end we have a slanted line. To add two vectors, you put one at the end of the other, the sum of these two vectors is then the vector coming from the origin of the first and arriving at the end of the second one.
sum of vectors

You can multiply a vector by a number. If you multiply a vector by 3 you get a vector 3 times longer :
multiplication

If you multiply a vector by a negative number it changes its direction, for example if you multiply vector by -2 :
vector

You can also multiply two vectors by each other but to understand that, you have to understand the 10th Grade. It is the dot product.

Finally, you agree that 8 - 5 = 8 + (-5). To subtract two vectors, you add the first one with the opposite of the second one, that is to say the second one that you multiplied by (-1).
subtraction of vectors

Coordinates of one point

With two "perpendicular" vectors with the same origin and the same length, you can locate points in a plan thanks to coordinates.
math

To go from 0 to A, you must add twice the vector coordinates of a point with 5 times the vector coordinates of a point. We say that 2 is the abscissa of A and that 5 is the ordinate of A. We write: A(2;5), which reads "point A whose coordinates are 2, 5". This technique enables us to locate with numbers the position of all points of such a drawing.

Distance between 2 points in a frame

If A(2,5) and B(4,1), you can calculate the distance from A to B.

lengths in a map

If you use Pythagoras' theorem, the result is :
lengths in a map
In general, to calculate the distance between two points when you know their coordinates, if and lengths in a map, then :

lengths in a map

You must learn this formula.


Coordinates of the middle of a segment

Otherwise, if you are asked to find the coordinates of the middle of a segment [AB], you must use the two formulas to find the abscissa and the ordinate of the middle M of the segment. You must calculate the mean of the coordinates of A and B :
math
math

Coordinates of a vector

The coordinates of a vector are the numbers saying how much the vector will go towards the right and how much it will go upwards. To calculate the coordinates of a vector, you must use the formula :
coordinates


Finally, letís come back to linear functions, we saw that they could be represented with a graph. Before skipping to ninth Grade, you must also know that if coordinates and coordinates are two point on the straight line representing the linear function, then the number slope of a line is the slope of the linear function. You can also say that it is the slope of the straight line.



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Vectors

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