Question:
If $\vec{a}$ is a nonzero vector of magnitude ' $a$ ' and $\lambda$ a nonzero scalar, then $\lambda \vec{a}$ is unit vector if
(A) $\lambda=1$
(B) $\lambda=-1$ (C) $a=|\lambda|$
(D) $a=\frac{1}{|\lambda|}$
Solution:
Vector $\lambda \vec{a}$ is a unit vector if $|\lambda \vec{a}|=1$.
Now,
$|\lambda \vec{a}|=1$
$\Rightarrow|\lambda||\vec{a}|=1$
$\Rightarrow|\vec{a}|=\frac{1}{|\lambda|}$ $[\lambda \neq 0]$
$\Rightarrow a=\frac{1}{|\lambda|}$ $[|\vec{a}|=a]$
Hence, vector $\lambda \vec{a}$ is a unit vector if $a=\frac{1}{|\lambda|}$.
The correct answer is D.
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