Porisma

Worked solution

Walk a fixed distance in a vector’s direction

Alice walks 5 units in the direction of p\mathbf p. She then walks 3 units in the direction of q\mathbf q. What is the expression of the resultant vector with reference to her starting point? (p\mathbf p and q\mathbf q are non-parallel vectors.)

(a) 5p^+3q^5\widehat{\mathbf p}+3\widehat{\mathbf q}; (b) 5p+3q5\mathbf p+3\mathbf q; (c) no difference.

CJC · Vectors Basic · Teacher poll

Answer

(a) 5p^+3q^5\widehat{\mathbf p}+3\widehat{\mathbf q}.

Compare your reasoning

Normalise each direction before scaling

r=5p∣p∣+3q∣q∣\mathbf r=5\frac{\mathbf p}{\lvert\mathbf p\rvert}+3\frac{\mathbf q}{\lvert\mathbf q\rvert}

A direction vector need not have magnitude 1. The fixed distances 5 and 3 multiply unit directions. Both direction vectors are nonzero.

How did this question go?

Common mistakes

5p+3q5\mathbf p+3\mathbf q would have leg lengths 5∣p∣5\lvert\mathbf p\rvert and 3∣q∣3\lvert\mathbf q\rvert. With nonparallel directions it agrees with (a) only when both given vectors are unit vectors.

Solution notes

The Teacher supplies the poll prompt and all three choices, without a printed answer key. This explanation is authored; the page reference identifies that source prompt.