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

• Vectors can be defined in any number of dimensions, though we focus here only on 3-space.

• When drawing a vector in 3-space, where you position the vector is unimportant; the vector's essential properties are just its magnitude and its direction. Two vectors are equal if and only if corresponding components are equal.

• A vector of norm 1 is called a unit vector. The coordinate vectors are examples of unit vectors.

• The zero vector, 0 = (0,0,0), is the only vector with magnitude 0.

Basic Operations on Vectors

To add or subtract vectors u = (u₁,u₂,u₃) and v = (v₁,v₂,v₃),

add or subtract the corresponding coordinates:

To mulitply vector u by a scalar k, multiply each coordinate of u by k:

Example

The 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 ¹ 0, we can scale (or normalize) v to the unit vector v/ ||v|| pointing in the same direction as v.

Dot Product

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:

u · v = u1v1+u2v2+u3v3 u · v is a number = ì í î || u || || v || cosq if u ¹ 0, v ¹ 0 0 if u = 0 or v = 0       where 0 £ q £ p is the angle between u and v

• u · v = v · u

• u · (v + w) = (u · v) + (u · w)

• u · u = || u ||²

Example

If u = (1,-2,2) and v = (-4,0,2), then

u · v	=	(1)(-4)+(-2)(0)+(2)(2)
	=	-4+0+4
	=	0

Using the second definition of the dot product with || u || = 3 and || v || = 2Ö5,

Though we might not have guessed it, u and v are perpendicular to each other!

In general,

Projection of a Vector

It 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||cosq, where 0 £ q £ p is the angle between u and v. Let's normalize u to u/|| u || and then scale this by the magnitude || v ||cosq:

Projection of v onto u	= (|| v || cosq) u ||u||
	= ||v|| ||u|| cosq ||u||2 u
	= v · u ||u||2 u

The perpendicular vector component of v is then just the difference between v and the projection of v onto u.

In summary,

projection of v onto u: = v · u ||u||2 u vector component of v perpendicular to u: = v - v · u ||u||2 u

Cross Product

Let u = (u₁,u₂,u₃) and v = (v₁,v₂,v₃). The cross 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 £ q £ p 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,

• u × v = - ( v × u)

• u × ( v + w ) = (u × v ) + ( u × w )

• u × u = 0

In the following Exploration, select values for the components of u and v. You will see u · v and u × v computed and u, v, and u × v displayed on a coordinate system.

Let u = (u₁,u₂,u₃) and v = (v₁,v₂,v₃).

• Basic Operations, Norm of a vector

  u + v = (u1+v1,u2+v2,u3+v3) u - v = (u1-v1,u2-v2,u3-v3) k u = (ku1,ku2,ku3) || v || = Ö x2+y2+z2

• Dot Product

  u · v = u1v1+u2v2+u3v3 = ì í î || u || || v || cosq if u ¹ 0, v ¹ 0 0 u = 0 or v = 0       where 0 £ q £ p is the angle between u and v

  For u ¹ 0, v ¹ 0, u · v = 0 if and only if u is orthogonal to v.

• Projection of a Vector

  projection of v onto u:	= v · u ||u||2 u
  vector component of v perpendicular to u:	= v - v · u ||u||2 u

• Cross Product
  u × v = (u₂v₃-u₃v₂)i - (u₁v₃-u₃v₁)j + (u₁v₂-u₂v₁)k

  || u × v || = || u || || v || sinq for u ¹ 0, for v ¹ 0

  where 0 £ q £ p is the angle between u and v.

  For u ¹ 0, v ¹ 0, u × v = 0 if and only if u is parallel to v.