If bar a = 3i + j + k, (i) Find the magnitude of bar a (ii) if the projection of bar a on another vector bar b is √14

Question

If \(\bar a\) = 3i + j + 2k,

(i) Find the magnitude of \(\bar a\).

(ii) If the projection of \(\bar a\) on another vector \(\bar b\) is \(\sqrt{14}\), which among the following could be \(\bar b\) ?

(a) i + j + k

(b) 6i + 2j + 4k

(c) 3i – j + 2k

(d) 2i + 3j + k

(iii) If \(\bar a\) makes an angle 60° with a vector \(\bar c\), find the projection of \(\bar a\) on \(\bar c\)

Answer 1

(i) |\(\bar a\)| = |3i + j + 2k| = \(\sqrt{14}\)

(ii) Since projection of \(\bar a\) on another vector \(\bar b\) and magnitude of \(\bar a\) is \(\sqrt{14}\), then \(\bar a\) and \(\bar b\) are parallel,

(b) 6i + 2j + 4k.

(iii) Projection of \(\bar a\) on \(\bar c\)

= |\(\bar a\)|cos60° = \(\sqrt{14} \times \frac{1}{2}\) = \(\frac{\sqrt{14}}{2}.\)