Consider the vectors p=2i^−j^+4k^ and q=−i^+3j^+2k^.
(i) State the algebraic (component) formula for the scalar product of two vectors expressed in the Cartesian basis and explain in not more than two sentences why the mutual perpendicularity of i^,j^,k^ justifies the formula.
(ii) Prove that the scalar product is distributive over vector addition, that is,
(a+b)⋅c=a⋅c+b⋅c for any vectors a,b,c.
(iii) Verify the result obtained in part (ii) for the vectors
p,q and r=3i^+2j^−5k^ by evaluating (p+q)⋅r and comparing it with p⋅r+q⋅r. State your conclusion clearly.