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I. Different expressions of the scalar product :
1. Colinear vectors :
be and
two non-zero collinear vectors, such that
and
.
– If and
have the same meaning:
.
– If and
are in opposite directions:
.
– If or
then
.
– is the scalar square of the vector
2. Any vectors :
Let and
be two non-zero vectors such that
and
.
So:
.
A’ and B’ are respectively the orthogonal projects of A on (OB) and of B on (OA).
3. Properties :
Let (x;y) and (x’;y’) be the respective coordinates of the vectors and
in any orthonormal reference frame.
.
II. Scalar product and orthogonality :
To say that and
are two orthogonal vectors means that :
– Either or
;
– Let (OA)(OB), with
and
nonzero.
2. Property:
.
III. Properties of the scalar product :
Properties:
Let be three vectors and k a real number.
– (symmetry).
– (linearity)
– (linearity)
– (linearity)
– (remarkable identity)
– (remarkable identity)
– (remarkable identity)
IV. Applications of the scalar product:
1. Scalar product and cosine :
Let and
be non-zero.
2. Al-Kashi theorem:
Let ABC be a triangle such that AB=c, AC=b and BC=a.
We have:
3. Median theorem:
Let A and B be two distinct points and I the middle of the segment [AB].
For any point M, :
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