10th Grade - Dot product

10th Grade lesson

8 - Dot product

Dot product

The product of two vectors is a number. We call it the dot product, or scalar product, of these two vectors. Put the origin of the vectors at the same place.

dessin definition du dot product
definition du dot product

The dot product of vectors vector AB and vector AC is the length AB multiplied by the length AH. So it depends on the norm (the length) of the two vectors and on the angle that is between them. If the two vectors form a right angle then the point H is the same than A and the dot product is zero. We say that the 2 vectors are orthogonal if their dot product is zero. If x is the angle between the two vectors, then :
dot product calculation
hence dot product calculation
hence dot product calculation

dot product formula

If you know the coordinates of the vectors and if vector coordinates and vector coordinates, hence dot product.

If dot product calculation example and dot product calculation example, dot product calculation example then the two vectors are not orthogonal.

Al Kashi's theorem

First you have to know that the calculation with the dot product is the same than with a normal product, and you have to notice that vector so that vector.

Hence :
al-kashi theorem proof

The following relationship is known as Al Kashi's theorem :
al-kashi theorem
If A is a right angle, it gives us Pythagorean theorem.

>>> Transformations of the plane >>>

Dot product

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