The distance of the point $$($$−$$1$$,−$$5$$,−$$10)$$ from the point of intersection of the line r¯$$=2i$$¯−j¯$$+2k$$¯$$+$$λ(3i$$¯$$+4j$$¯$$+2k$$¯$$) and plane r¯.$$(i$$¯−j¯$$+k$$¯$$)=5$$ is

Equation of line x−$$23=y+14=z$$−$$22$$ a point on the line it lies on the plane $$x-y+z=5p(2$$,−$$1$$,$$2)Q($$−$$1$$,−$$5$$,−$$10)PQ=9$$−$$16+144=13$$

Planes in 3D

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Question

The distance of the point $(- 1, - 5, - 10)$ from the point of intersection of the line $\bar{r} = 2 \bar{i} - \bar{j} + 2 \bar{k} + λ (3 \bar{i} + 4 \bar{j} + 2 \bar{k})$ and plane $\bar{r} . (\bar{i} - \bar{j} + \bar{k}) = 5$ is

Moderate

Solution

Equation of line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}$
a point on the line it lies on the plane $x - y + z = 5$
$p (2, - 1, 2) Q (- 1, - 5, - 10)$
$P Q = \sqrt{9 - 16 + 144} = 13$

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