In order to measure many physical quantities, such as force or velocity, we need to determine both a magnitude and a direction. Such quantities are conveniently represented as vectors. The direction of a vector v in 3-space is specified by its components in the x, y, and z directions, respectively:
where i, j, and k are the coordinate vectors along the x-, y-, and z-axes. The magnitude of a vector v = (x,y,z), also called its length or norm, is given by
Notes
Basic Operations on VectorsTo add or subtract vectors u = (u1,u2,u3) and v = (v1,v2,v3), add or subtract the corresponding coordinates:
u - v = (u1-v1,u2-v2,u3-v3) To mulitply vector u by a scalar k, multiply each coordinate of u by k:
ExampleThe vector v = (2,1,-2) = 2i + j - 2k has magnitude
Thus, the vector 1/3v = (2/3,1/3,-2/3) is a unit vector in the same direction as v. In general, for v ≠ 0, we can scale (or normalize) v to the unit vector v/ ||v|| pointing in the same direction as v.
Dot Product (prodotto scalare)Let u = (u1,u2,u3) and v = (v1,v2,v3). The dot product u · v (also called the scalar procuct or Euclidean inner product) of u and v is defined in two distinct (though equivalent) ways:
Properties of the Dot Product
ExampleIf u = (1,-2,2) and v = (-4,0,2), then
Using the second definition of the dot product with || u || = 3 and || v || = 2√5,
Though we might not have guessed it, u and v are perpendicular to each other! In general,
Two non-zero vectors u and v are perpendicular (or orthogonal) if and only if u ·v = 0.
Projection of a VectorIt is often useful to resolve a vector v into the sum of vector components parallel and perpendicular to a vector u. Consider first the parallel component, which is called the projection of v onto u. This projection should be in the direction of u and should have magnitude ||v||cosθ, where 0 ≤ θ ≤ π is the angle between u and v. Let's normalize u to u/|| u || and then scale this by the magnitude || v ||cosθ:
The perpendicular vector component of v is then just the difference between v and the projection of v onto u. In summary,
Cross Product (prodotto vettoriale)Let u = (u1,u2,u3) and v = (v1,v2,v3). The cross product (also called the vector product) u × v yields a vector perpendicular to both u and v with direction determined by the right-hand rule. Specifically,
It can also be shown that
where 0 ≤ θ ≤ π is the angle between u and v.
Thus, the magnitude || u ×v || gives the area of the parallelogram formed by u and v. As implied by the geometric interpretation,
Non zero vectors u and v are parallel if and only if u × v = 0.
Properties of the Cross Product
Let u = (u1,u2,u3) and v = (v1,v2,v3).
|