The direction ratios of a bisector of the ang | vedclass

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Question

(A) Let the vectors along the two lines be $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = \sqrt{3}\hat{i} - \sqrt{3}\hat{j} + 0\hat{k}$.First

Vedclass · Accepted Answer

$1+\sqrt{3}, 1-\sqrt{3}, 2$

Vedclass · Answer

(A) Let the vectors along the two lines be $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = \sqrt{3}\hat{i} - \sqrt{3}\hat{j} + 0\hat{k}$.First, calculate the magnitudes of the vectors:$|\vec{a}| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}$.$|\vec{b}| = \sqrt{(\sqrt{3})^2 + (-\sqrt{3})^2 + 0^2} = \sqrt{3 + 3 + 0} = \sqrt{6}$.Since the magnitudes are equal $(|\vec{a}| = |\vec{b}|)$, the direction ratios of the angle bisector are given by the components of the vector $\vec{a} + \vec{b}$ or $\vec{a} - \vec{b}$.Calculating $\vec{a} + \vec{b} = (1 + \sqrt{3})\hat{i} + (1 - \sqrt{3})\hat{j} + (2 + 0)\hat{k}$.Thus, the direction ratios are $(1 + \sqrt{3}, 1 - \sqrt{3}, 2)$.Comparing this with the given options, option $A$ is correct.

The direction ratios of a bisector of the angle between two lines whose direction ratios are $1, 1, 2$ and $\sqrt{3}, -\sqrt{3}, 0$ are

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