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(A) 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} + \lambda(\v

Vedclass · Accepted Answer

$(\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$

Vedclass · Answer

(A) The vector equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is given by $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$.Here,the position vectors of points $P(1, 2, 3)$ and $Q(2, 3, 4)$ are $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k}$.The direction vector of the line is $\vec{b} - \vec{a} = (2 - 1)\hat{i} + (3 - 2)\hat{j} + (4 - 3)\hat{k} = \hat{i} + \hat{j} + \hat{k}$.Thus,the vector equation of the line is $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$.

The vector equation of the line passing through $P(1, 2, 3)$ and $Q(2, 3, 4)$ is

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