Scalar product

The scalar or dot product of two vectors, u and v, is the real number: u·v = u1·v1 + u2·v2 + u3v3

Properties:

- Commutative property: u·v = v·u

- Associative with respect to the multiplication by a scalar: (k·uv = k·(u·v)    k€R

- Distributive property: u·(v + w) = u·v + u·w

-

- v·v = |v|2

- Cauchy-Schwartz inequality: |u·v| ≤ |u|·|v|

Theorem: if θ (≤ π) is the angle between u and v, then:

Demonstration:

Cosine Theorem→ |u-v|2=|u|2+|v|2-2|u||v|cosθ

On the other hand: (u-v)2=(u-v)(u-v)=u2-uv-vu+v2=|u|2+|v|2-2uv

Joining: -2uv =-2|u||v|cosθ

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 non-zero vectors. If w=w1+w2 with w1 parallel to v and w2 orthogonal to v, we say that w1 is the orthogonal projection of w over v and w2 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 right-angled in A.

Solutions: 1.- a) 70o53'36''; b) (3/7,-9/14,3/4); 2.- k =  0 or k = 1

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