Definitions.
  • A two-dimensional vector is an ordered pair of real numbers v = (a, b).
  • a and b are called the components of v.
  • The length or magnitude of the vector v = (a, b) is

  • A three-dimensional vector is an ordered triple of real numbers w = (a, b, c).
  • a, b and c are called the components of w.
  • The length or magnitude of the vector w = (a, b, c) is

  • A scalar is a real number.

Discussion. [Using Flash] [Using Java]

Definition. Suppose that v = (a, b) and w = (c, d) are two vectors. The sum of these two vectors is defined to be the vector

v + w = (a + c, b + d).

Similarly, if v = (a, b, c) and w = (d, e, f) then the sum

v + w = (a + d, b + e, c+ f).

Discussion. [Using Flash] [Using Java]

Definition. Suppose that c is a scalar and v = (a, b) is a vector, then scalar multiplication is defined by
c v = c (a, b) = (c a, c b)

Discussion. [Using Flash] [Using Java]

Definition. Suppose that v and w are two vectors. The difference of these two vectors is defined to be the vector

v - w = v + (-1)w

where (-1)w is the scalar multiplication of -1 and w.

Often, we write - w instead of (-1)w.

The zero vector, 0, is the vector all of which components are zero.

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Theorem. Suppose that u, v and w are vectors and suppose that a and b are scalars. Then
  • u + v = v + u (the commutative law of addition)
  • u + (v + w) = (u + v) + w (the associative law of addition)
  • u + 0 = u
  • u + (-u) = 0 (existence of additive inverses)
  • a(bu) = (ab)u
  • (a + b)u = au + bu
  • a(u + v) = au + av
  • 1u = u

Discussion. [Using Flash] [Using Java]

Definition. The standard unit 2-dimensional vectors are

i = (1,0) and j = (0,1).

The standard unit 3-dimensional vectors are

i = (1,0,0), j = (0,1,0) and k = (0,0,1).

Discussion. [Using Flash] [Using Java]

Problems.