Let be a fixed non-zero vector and let be a fixed real number

MATHEMATICS · Multi-part · CBSE · Mathematics · Vector Algebra · Vector Algebra

Problem

Let

\vec a

be a fixed non-zero vector and let

k>0

be a fixed real number. Consider all vectors

\vec b

having magnitude

k

. For such vectors define

P=\dfrac{\vec a \cdot \vec b}{k},

which equals the length of the projection of

\vec a

\vec b

. (i) Show that

P

attains a maximum when

\vec b

is parallel to

\vec a

and a minimum when

\vec b

is antiparallel to

\vec a

. (ii) Obtain the maximum and minimum possible numerical values of

P

in terms of

|\vec a|

and

k

. (iii) Take

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

and

k = 7

. Find vectors

\vec b_{\text{max}}

and

\vec b_{\text{min}}

(in component form) that realise the extreme values obtained in part (ii).