Which of the following vectors is equally inc | vedclass

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Question

(D) vector $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$ is equally inclined to the coordinate axes if its direction cosines are equal,i.e.,$\cos \alpha

Vedclass · Accepted Answer

$4 \hat{i}+4 \hat{j}+4 \hat{k}$

Vedclass · Answer

(D) vector $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$ is equally inclined to the coordinate axes if its direction cosines are equal,i.e.,$\cos \alpha = \cos \beta = \cos \gamma$. This implies that the direction ratios must be equal in magnitude,i.e.,$|a| = |b| = |c|$. Checking the options: For option $D$,$\vec{v} = 4\hat{i} + 4\hat{j} + 4\hat{k}$. The direction ratios are $a=4, b=4, c=4$. The magnitude is $|\vec{v}| = \sqrt{4^2 + 4^2 + 4^2} = \sqrt{48} = 4\sqrt{3}$. The direction cosines are $\left(\frac{4}{4\sqrt{3}}, \frac{4}{4\sqrt{3}}, \frac{4}{4\sqrt{3}}\right) = \left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$. Since all direction cosines are equal,the vector $4\hat{i} + 4\hat{j} + 4\hat{k}$ is equally inclined with the coordinate axes.

Which of the following vectors is equally inclined with the coordinate axes?

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