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Question

(C) The vector equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is given by $\vec{r} = \vec{a} + t(\vec{b}

Vedclass · Accepted Answer

$a_1(1 - t)\hat{i} + a_2(1 - t)\hat{j} + a_3(1 - t)\hat{k} + (b_1\hat{i} + b_2\hat{j} + b_3\hat{k})t$

Vedclass · Answer

(C) The vector equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is given by $\vec{r} = \vec{a} + t(\vec{b} - \vec{a})$,where $t$ is a scalar parameter.Expanding this,we get $\vec{r} = \vec{a} + t\vec{b} - t\vec{a} = (1 - t)\vec{a} + t\vec{b}$.Substituting $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ into the equation:$\vec{r} = (1 - t)(a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) + t(b_1\hat{i} + b_2\hat{j} + b_3\hat{k})$$\vec{r} = a_1(1 - t)\hat{i} + a_2(1 - t)\hat{j} + a_3(1 - t)\hat{k} + t(b_1\hat{i} + b_2\hat{j} + b_3\hat{k})$.Thus,option $C$ is correct.

The equation of the line passing through the points $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ is given by:

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