First slide

Straight lines in 3D

Question

$A line passes through the point A (\hat{i} + 2 \hat{j} + 3 \hat{k}) and is parallel to the vector \vec{V} (\hat{i} + \hat{j} + \hat{k}) .$ The shortest distance from the origin, to the line is……

Moderate

A

$\sqrt{2}$
B

$\sqrt{4}$
C

$\sqrt{5}$
D

$\sqrt{6}$

Solution

$Equation of line: \frac{x - 1}{1} = \frac{y - 2}{1} = \frac{z - 3}{1} = t .$
$Let P = (t + 1, t + 2, t + 3) be the foot of the perpendicular from origin$ $Then (t + 1) (1) + (t + 2) (1) + (t + 3) (1) = 0$
$t = - 2$
$∴ P = (- 1, 0, 1) a n d O P = \sqrt{2}$

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