MHT CET · Maths · Vector Algebra

If vector $\vec{r}$ with direction cosine $l, m, n$ is equally inclined to the co-ordinate axes, then the total number of such vectors is

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Correct Answer

(C) 8

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Detailed explanation

$\bar{r}=|\bar{r}|\left( \pm \frac{1}{\sqrt{3}} \hat{i} \pm \frac{1}{\sqrt{3}} \hat{j} \pm \frac{1}{\sqrt{3}} \hat{k}\right)$
For equally inclined to co-ordinate axes.
$\Rightarrow|l|=|m|=|n|$
$\Rightarrow l^2+m^2+n^2=1$
$\Rightarrow 3 l^2=1$
$\Rightarrow l^2=\frac{1}{3}$
$\Rightarrow l= \pm \frac{1}{\sqrt{ } 3}=m=n \Rightarrow l, m, n$ each has 2 choices.
$\therefore $ Total lines $=2^3$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If vector r → with direction cosine l,m,n is equally inclined to the co-ordinate axes, then the total number of such vectors is

Step-by-step Solution

If vector $\vec{r}$ with direction cosine $l, m, n$ is equally inclined to the co-ordinate axes, then the total number of such vectors is