VectorsInSpace
Scalar product
The scalar or dot product of two vectors, u and v, is the real number: u·v = u_{1}·v_{1} + u_{2}·v_{2} + u_{3}v_{3}
Properties:
 Commutative property: u·v = v·u
 Associative with respect to the multiplication by a scalar: (k·u)·v = k·(u·v) k€R
 Distributive property: u·(v + w) = u·v + u·w

 v·v = v^{2}
 CauchySchwartz inequality: u·v ≤ u·v
Theorem: if θ (≤ π) is the angle between u and v, then:
Demonstration:
Cosine Theorem→ uv^{2}=u^{2}+v^{2}2uvcosθ
On the other hand: (uv)^{2}=(uv)(uv)=u^{2}uvvu+v^{2}=u^{2}+v^{2}2uv
Joining: 2uv =2uvcosθ
Then
Two vectors, u and v, are said to be orthogonal if u·v=0
Properties:
–If u = 0 → u is orthogonal to every vector
–If u,v ≠ 0, then: u·v = 0 ↔ θ = π/2
An orthonormal basis is formed by unit vectors that are orthogonal to each other.
For example, the canonical basis is an orthonormal basis.
Let v and w two nonzero vectors. If w=w_{1}+w_{2} with w_{1} parallel to v and w_{2} orthogonal to v, we say that w_{1} is the orthogonal projection of w over v and w_{2} its orthogonal component.
We name:
Then:
Demonstration:
NOTE: If u is a unit vector
Then, the scalar product by a unit vector, u, measures the length of the orthogonal projection in the direction of u.
Exercises:
1. Let u(1,1,2) and v(2,3,1):
a) Calculate the angle that u and v form.
b) Calculate the orthogonal projection of u over v.
2. Let the points A(1,5,k), B(3,k,1), C(k,5,2) the vertices of a triangle. Find the value of k that makes the triangle rightangled in A.
Solutions: 1. a) 70^{o}53'36''; b) (3/7,9/14,3/4); 2. k = 0 or k = 1
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