9.11 Angle between a line and a plane

Angle between a line and a plane 

To find the angle between a straight line and a two dimensional plane, we need to find another appropriate line on the plane so we can use the above method for finding the angle. The line on the plane should start at the same point at the original line, and pass through a point on the plane that is vertically below the end point on the original line. 

Example 1

Find the angle between the line $CD$CD   and the base plane of the following square pyramid: 

 

 

 

 

 

 

 

 

Think: The point directly below point $D$D  on the base plane is the point $(2,2,0)$(2,2,0) which we can call $X$X .   So we actually need to find the angle between the two line segments $CD$CD and $CX$CX:

 

Do: Now we must find the lengths of both line segments, starting with $CD$CD:

$CD$CD	$=$=	$\sqrt{(2-0)^2+(2-4)^2+(4-0)^2}$√(2−0)2+(2−4)2+(4−0)2
	$=$=	$\sqrt{4+4+16}$√4+4+16
	$=$=	$\sqrt{24}$√24

 

Now $CX$CX:

$CX$CX	$=$=	$\sqrt{(2-0)^2+(2-4)^2+(0-0)^2}$√(2−0)2+(2−4)2+(0−0)2
	$=$=	$\sqrt{4+4}$√4+4
	$=$=	$\sqrt{8}$√8

 

Now we can can find the angle using the formula: 

$\tan\theta$tanθ	$=$=	$\frac{\text{rise}}{\text{run}}$riserun​
	$=$=	$\frac{\sqrt{24}}{\sqrt{8}}$√24√8​
$\theta$θ	$=$=	$\arctan\left(\frac{\sqrt{24}}{\sqrt{8}}\right)$arctan(√24√8​)
	$=$=	$60^\circ$60°