Coordinate System 2

Figure 2 
              Drawing for Example 1.

If the coordinates of point P are ( x, y),

An algebraic vector is an ordered pair of real numbers. An algebraic vector that corresponds to standard geometric vector  is denoted as ⟨ a, b⟩ if terminal point P has coordinates of (a, b). The numbers a and b are called the components of vector ⟨ a, b⟩ (see Figure 3 ).

Figure 3 
              Components of a vector.

If a, b, c, and d are all real numbers such that a = c and b = d, then vector v = ⟨ a, b⟩ and vector u = ⟨ c, d⟩ are said to be equal. That is, algebraic vectors with equal corresponding components are equal. If both components of a vector are equal to zero, the vector is said to be the zero vector. The magnitude of a vector v = ⟨a, b⟩ is .

Example 2: What is the magnitude of vector u = ⟨3, −5⟩?

Vector addition is defined as adding corresponding components of vectors—that is, if v = ⟨ a, b⟩ and u = ⟨c, d⟩, then v + u = ⟨a + c, b + d⟩ (Figure 4 ).

Figure 4 
              Vector addition.

Scalar multiplication is defined as multiplying each component by a constant—that is, if v = ⟨a, b⟩ and q is a constant, then q v = q⟨a, b⟩ = ⟨qa, qb⟩.

Example 3: If v = ⟨8, −2⟩ and w = ⟨3, 7⟩ then find 5 v −2 w.

A unit vector is a vector whose magnitude is 1. A unit vector v with the same direction as a nonzero vector u can be found as follows: