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Worked solution

Interpret parametric lines and zero cross products

(a) Interpret geometrically the vector equation r=a+tb\mathbf r=\mathbf a+t\mathbf b, where a\mathbf a and b\mathbf b are constant vectors and tt is a parameter.

(b) The points PP, QQ and RR have position vectors p\mathbf p, q\mathbf q and r\mathbf r respectively. The points PP and QQ are fixed and RR varies. Given that p\mathbf p is non-zero and (r−p)×q=0(\mathbf r-\mathbf p)\times\mathbf q=\mathbf0, describe geometrically the set of all possible positions of the point RR.

CJC · Discussion · Question 8

Answer

(a) A line through a parallel to b if b is nonzero; otherwise one point. (b) If q is nonzero, the line r=p+λq\mathbf r=\mathbf p+\lambda\mathbf q; if q is zero, all space.

(a) Geometrical interpretation

For real t and b≠0\mathbf b\ne\mathbf 0, this is a line through the point with position vector a and parallel to b. If b=0\mathbf b=\mathbf 0, it represents the single point a.

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(b) Nonzero q

When q≠0\mathbf q\ne\mathbf 0, zero cross product means r−p=λq\mathbf r-\mathbf p=\lambda\mathbf q for some real λ\lambda. Thus

r=p+λq,λ∈R.\mathbf r=\mathbf p+\lambda\mathbf q,\quad\lambda\in\mathbb R.

R can be any point of the line through P parallel to OQ, including P itself when λ=0\lambda=0.

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(b) Zero-vector cases

If r−p=0\mathbf r-\mathbf p=\mathbf 0, then R=PR=P; this is already included above when q is nonzero. If q=0\mathbf q=\mathbf 0, the cross product vanishes for every r, so R is unrestricted in three-dimensional space. The given p≠0\mathbf p\ne\mathbf 0 does not exclude either case.

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Common mistakes

A zero cross product does not force a unique line when one factor is the zero vector.

Solution notes

The source main answer gives the line and its blue notes flag zero-vector exceptions. Its phrase “any line/plane” for q=0 is made precise as all space; it does not restrict R to a particular plane. The source extra assumption that r and r−p are nonzero is not in the printed prompt and is not imposed here.

Read the hint