How do I show perpendicularity and find the angle between a line and a plane?

[hex.halo]

Homework Statement

Find the line ... . Show that it is perpendicular to the plane A and find the angle that the line makes with the plane B

Homework Equations

The Attempt at a Solution

I've found the line, but how do I go about showing it's perpendicular and finding the angle?

 

[rohanprabhu]

The plane must be in either of the two forms.. either it's in a vector form, or Cartesian form. Let's say, it's in a cartesian form..

$$Ax + By + Cz + D = 0$$

So you have the direction ratios of the normal to the plane. for a line to be perpendicular to this, you need to get the direction ratios of the line as well. Once you have that, use the check for perpendicularity:

$$l_1 l_2 + m_1 m_2 + n_1 n_2 = 0$$

which is equivalent to checking if the dot product of the two vectors is zero or not, which i'd say is a better method.

For finding the angle, find a line parallel to the given line [using the direction ratios] and do the same thing for the plane's normal.. and then use the formula:

$$\cos{(\theta)} = \frac{\overrightarrow{a}~.~\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}$$