A student claims that whenever the scalar product of two vectors comes out to be exactly zero, the vectors must be perpendicular, and whenever the vector product of two vectors comes out to be exactly zero, the vectors must be parallel.
(i) Explain, with reference to the definitions of the two products, why each part of the student’s claim is generally true.
(ii) In a laboratory experiment the following measured vectors (in arbitrary units) were obtained:
p=1.000i^+2.000j^+3.000k^ and q=−2.000i^−4.000j^−6.000k^.
Their vector product was calculated as p×q=(2.0×10−12)i^−(1.0×10−12)j^+(4.0×10−13)k^.
Discuss whether these vectors should be regarded as practically parallel in the light of instrumental uncertainties.
(iii) Give one additional numerical example each to illustrate (a) a pair of vectors that are exactly perpendicular, and (b) a pair that are exactly parallel. For each example, present the calculation that confirms your claim.