VectorsA vector, it's this :
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 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 , is the point which is at the end of the vector when the origin of the vector is point A.
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.
You can multiply a vector by a number. If you multiply a vector by 3 you get a vector 3 times longer :
If you multiply a vector by a negative number it changes its direction, for example if you multiply by -2 :
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).
Coordinates of one pointWith two "perpendicular" vectors with the same origin and the same length, you can locate points in a plan thanks to coordinates.
To go from 0 to A, you must add twice the vector with 5 times the vector . 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 frameIf A(2,5) and B(4,1), you can calculate the distance from A to B.
If you use Pythagoras' theorem, the result is :
In general, to calculate the distance between two points when you know their coordinates, if and , then :
You must learn this formula.
Coordinates of the middle of a segmentOtherwise, 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 :
Coordinates of a vectorThe 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 :
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 and are two point on the straight line representing the linear function, then the number is the slope of the linear function. You can also say that it is the slope of the straight line.