Mathematics

Mathematics

Problem

Vectors

\vec{p}=2\mathbf{i}-3\mathbf{j}+\mathbf{k}

and

\vec{q}=\mathbf{i}+4\mathbf{j}-2\mathbf{k}

are fixed. A third vector is formed as

\vec{r}=\vec{p}+p\,\vec{q}

, where

p

is a real parameter. a. (i) Find the value of

p

for which

\vec{r}

is perpendicular to

\vec{p}-\vec{q}

. b. (ii) For the value of

p

obtained in part (i), calculate

|\vec{r}|

. c. (iii) Show that, for this value of

p

, the vectors

\vec{r}

and

\vec{q}

are not parallel.

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