A parallel line and a perpendicular plane

(-1,2,3)

Total marks: 8

Solution

(i)

Line parallel to a plane

$l$ passes through $A(-1,2,3)$ and $B(5,14,11)$ ; plane $p:2x+3y-6z=-7$ . Show $l\parallel p$ but $l\not\subset p$ .

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Hint

Direction $\overrightarrow{AB}=(6,12,8)\parallel(3,6,4)$ .
$\mathbf d\cdot\mathbf n=0$ .
Test whether $A$ is on $p$ .

Answer · (i)

u\cdot n=0

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A line above a plane

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

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(ii)

Distance from line to plane

Find the distance from $l$ to $p$ .

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Hint

Use any point on $l$ , for example $A$ .
Distance $=\frac{|\overrightarrow{CA}\cdot\mathbf n|}{|\mathbf n|}$ with $C$ on $p$ .

Answer · (ii)

1

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The shortest distance from A to the plane

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(iii)

Plane through a line, perpendicular to a plane

Find a Cartesian equation of the plane containing $l$ and perpendicular to $p$ .

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Hint

Its normal is perpendicular to both $\mathbf d=(3,6,4)$ and $\mathbf n=(2,3,-6)$ .

Answer · (iii)

-48x+26y-3z=91

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Two perpendicular planes

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A parallel line and a perpendicular plane

Solution

Line parallel to a plane

Direction

Parallel

$A$ not on $p$

A line above a plane

Distance from line to plane

A point on $p$

Vector

Distance

The shortest distance from A to the plane

Plane through a line, perpendicular to a plane

Normal

Through $A$

Equation

Two perpendicular planes

Direction

Parallel

AAA not on ppp

A point on ppp

Vector

Distance

Normal

Through AAA

Equation

$A$ not on $p$

A point on $p$

Through $A$