A line passing through the point A with position vector a = 4 i +… - Vidyadip

Question

A line passing through the point A with position vector $\overrightarrow{\text{a}} = 4\hat{\text{i}} + 2\hat{\text{j}} + 2\hat{\text{k}}$ is parallel to the vector$\overrightarrow{\text{b}} = 2\hat{\text{i}} + 3\hat{\text{j}} + 6\hat{\text{k}}.$ Find the length of the perpendicular drawn on this line from a point P with position vector$\overrightarrow{\text{r}_{1}} = \hat{\text{i}} + 2\hat{\text{j}} + 3\hat{\text{k}}.$

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Answer

$\overrightarrow{\text{r}} = (4\hat{\text{i}} + 2\hat{\text{j}} + 2\hat{\text{k}}) + \lambda(2\hat{\text{i}} + 3\hat{\text{j}} + 6\hat{\text{k}})\overrightarrow{\text{a}} + \lambda\overrightarrow{\text{b}}$
Let L be the foot of perpendicular
Position vector of List is $(2\lambda + 4)\hat{\text{i}} +( 3\lambda + 2) \hat{\text{j}} + (6\lambda + 2) \hat{\text{k}}$
$\overrightarrow{\text{PL}} = (2\lambda + 3)\hat{\text{i}} + 3\lambda \hat{\text{j}} + (6\lambda - 1)\hat{\text{k}}$
$\overrightarrow{\text{PL}}.\overrightarrow{\text{b}} = 2(2\lambda + 3)+3(3\lambda) + 6(6\lambda - 1) = 0$
$\Rightarrow\lambda = 0$
$\overrightarrow{\text{PL}} = 3\hat{\text{i}} - \hat{\text{k}}$
$\overrightarrow{\text{|PL|}} = \sqrt{10}$ units

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