A·eˆ 1 =A 1 A·ˆe 2 =A 2 A·ˆe 3 =A 3 (6 .10)represent respectively, the components or projections of the vector A onto the x, y
and z-axes. The projections A 1 , A 2 , A 3 of the vector A onto the coordinate axes
are scalars which are called the components of the vector A. From the definition of
the dot product of two vectors, the scalar components of the vector A satisfy the
equations
A 1 =A·ˆe 1 =|A|cos α, A 2 =A·ˆe 2 =|A|cos β, A 3 =A·eˆ 3 =|A|cos γ, (6 .11)where α, β, γ are respectively, the smaller angles between the vector A and the
x, y, z coordinate axes. The cosine of these angles are referred to as the direction
cosines of the vector A. These angles are illustrated in figure 6-7.
Figure 6-7. Unit vectors ˆe 1 ,ˆe 2 ,ˆe 3 and ˆeA= cos αˆe 1 + cos βˆe 2 + cos γˆe 3
The vector quantities
A 1 =A 1 ˆe 1 , A 2 =A 2 ˆe 2 , A 3 =A 3 ˆe 3 (6 .12)are called the vector components of the vector A. From the addition property of
vectors, the vector components of A may be added to obtain
A=A 1 ˆe 1 +A 2 ˆe 2 +A 3 ˆe 3 =|A|(cos αˆe 1 + cos βˆe 2 + cos γˆe 3 ) = |A|ˆeA (6 .13)This vector representation A =A 1 ˆe 1 +A 2 ˆe 2 +A 3 ˆe 3 is called the component form of
the vector A and the unit vector ˆeA= cos αˆe 1 + cos βˆe 2 + cos γeˆ 3 is a unit vector in
the direction of A.