Final answer:
The equation given represents the vector vec(a) broken down into its x, y, and z components along unit vectors hat(i), hat(j), and hat(k). The scalar components of vec(a) are the coefficients of these unit vectors, corresponding to their respective axes in the Cartesian coordinate system.
Step-by-step explanation:
The equation vec(a) = (vec(a)hat(i))hat(i) + (vec(a)hat(j))hat(j) + (vec(a)hat(k))hat(k) represents the expansion of any vector vec(a) into its components along the standard unit vectors in three-dimensional space, hat(i), hat(j), and hat(k). These unit vectors represent the x, y, and z axes, respectively, in Cartesian coordinates. The terms vec(a)hat(i), vec(a)hat(j), and vec(a)hat(k) represent the scalar components of the vector vec(a) along these axes.
In the Cartesian coordinate system, scalar products of unit vectors with themselves are always one, and with each other are zero, due to their orthogonality. Using these properties, we can say that the dot product of any vector with a unit vector gives the magnitude of that vector along the axis represented by the unit vector. For instance, Ax = vec(a) · hat(i) would represent the x-component of vector vec(a).
Therefore, the scalar components of the resultant vector are simply the coefficients of the unit vectors hat(i), hat(j), and hat(k) in the given vector representation.