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0593_C02_fm Page 23 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 23

where n , n , and n are mutually perpendicular unit vectors, then v , v , and v are called
y
z
x
z
x
y
the scalar components of V relative to n , n , and n . Then, from Eq. (2.4.10), the magnitude
z
y
x
of V is:
/
V = v 2 + v 2 y + v 2 z ) 12 (2.4.23)
( x
Observe that if V is zero, then V is zero, and each of the scalar components is also
zero. This is the basis for force equilibrium procedures of elementary mechanics.

2.5 Angle Between Two Vectors
The concept of the angle between two vectors is useful in developing the procedures of
vector multiplication. We already used this idea in Section 2.3 with the law of cosines (see
Figure 2.3.5). The angle between two vectors is defined as follows: Let A and B be any
nonzero vectors as in Figure 2.5.1. Let the vectors be connected tail to tail, as in Figure
2.5.2. Then the angle as shown is defined as the angle between the vectors. Observe that
θ always has values between 0 and 180˚.

2.6 Vector Multiplication: Scalar Product
Multiplying vectors can be accomplished in several ways, which we will review in this
and the following sections. Consider first the scalar product, so called because the result
is a scalar: Given any two vectors A and B, the scalar product, written as A · B, is defined as:

AB = A B cosθ (2.6.1)
⋅

where θ is the angle between A and B (see Section 2.5.1). Because a dot is placed between
the vectors, the operation is often called the dot product.
Observe in the definition of Eq. (2.6.1) that, if we interchange the positions of A and B,
the result remains the same. That is,

⋅
⋅
AB = A B cosθ = B A cosθ = B A (2.6.2)
Hence, the scalar product is commutative.
Consider some special cases. First, observe that the scalar product of two perpendicular
vectors is zero, as the cosθ is zero. Next, consider the scalar product of a vector A with itself:

⋅
AA = A A cosθ (2.6.3)