Question A.
A1. Consider the set of complex numbers. Do they form a vector space over the real numbers with v.a. and s.m. defined as the usual sum and product of complex numbers? Prove your answer.
A2. Consider the set of the rational numbers. Do they form a vector space over the real numbers if v.a. and s.m. are defined as the usual sum and product of real numbers?
Prove your answer.
(v.a. = vector addition; s.m. = scalar multiplication)
Question B. A body is subject to a force, φ, of 10 N along the Ox axis as well as to a wench, ψ. Find the resultant, φ + ψ, when ψ has intensity 10 N and is:
B1. A force with line of action the Oy axis.
B2. A wrench with pitch 1 m and screw axis along Oy.
B3. A force with line of action through point (0, 0, 1) directed as Oy. B4. A wrench with pitch 1 and axis through point (0, 0, 1) directed as Oy.
In each case, give the intensity and describe the screw, i.e., pitch and axis (or direction if the pitch is infinite), of the resultant wrench and illustrate it with a drawing. To describe an axis (a line), specify either: a point and a direction, {𝑞⃗ + 𝜆𝑢⃗⃗|𝜆 ∈ ℝ}; or two points.
Question C. The directed axis l passes from point (0, 0,1,) to point (1,1, 0). Find the Plücker coordinates of these wrenches:
C1. A unit force along l.
C2. A unit force against l.
C3. A unit couple with moment directed as l. C4. A wrench on l with pitch –1/2 m and intensity 4 N.
