Find the vector sum d = a - b, where a = 3i + 4j + (-1)k and b = 1i + (-3)j + (-2)k.

These vectors are shown in the diagram, along with -b, which is a negation and translation of b so that its tail coincides with the head of a. Vector d is drawn from the tail of a to the head of -b, as required by the definition of the vector sum.

To find the sum analytically, we subtract the x-components of the vectors to find the x-

component of the difference, and do the same for the y- and z-components:

c = (a_x - b_x)i + (a_y - b_y)j + (a_z - b_z)k

= (3 - 1)i + (4 - (-3))j +(-1 - (-2))k

= 2i + 7j +k