Vector planes

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October 12th 2008, 04:55 AM
free_to_fly

Vector planes

Q: What are the properties of the plane $\ b.(r - a) = 0 \$ ?
If O is the origin and a point C has position vector c, find:
a) the length of the projection of OC onto the plane
b) the distance of C from the plane.

For part a) this sound like it has something to do with the scalar product?
For part b) I know how to find the distance of a point to a plane using the formula:
$\ d = \left| {\frac{{n_1 a + n_2 b + n_3 c + d}}{{\sqrt {n_1^2 + n_2^2 + n_3^2 } }}} \right| \$

Help would be hugely appreciated.
October 12th 2008, 05:33 AM
earboth

Quote:

Originally Posted by free_to_fly

Q: What are the properties of the plane $\ b.(r - a) = 0 \$ ?
If O is the origin and a point C has position vector c, find:
a) the length of the projection of OC onto the plane
b) the distance of C from the plane.

For part a) this sound like it has something to do with the scalar product?
For part b) I know how to find the distance of a point to a plane using the formula:
$\ d = \left| {\frac{{n_1 a + n_2 b + n_3 c + d}}{{\sqrt {n_1^2 + n_2^2 + n_3^2 } }}} \right| \$

Help would be hugely appreciated.

I assume that you know that

$\vec b$ is the normal vector of the plane

$\vec r$ is the position vector of any point of the plane

$\vec a$ is the position vector of a fixed point which belongs to the plane.

To calculate the distance of a point from the plane you have to use Hesse's normal form of the plane:

$\vec b \cdot (\vec r - \vec a) = 0~\implies~\dfrac{\vec b}{|\vec b| } \cdot (\vec r - \vec a) = 0$

If you plug in the position vector of C instead of $\vec r$ you'll get the distance of C to the plane:

$d= \dfrac{\vec b}{|\vec b| } \cdot (\vec c - \vec a)$
October 12th 2008, 05:42 AM
free_to_fly

So when it asks me to find the length of the projection of OC onto the plane, is it asking me to find where OC intersects the plane and the distance between that point and the origin?
October 12th 2008, 10:49 AM
earboth

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Quote:

Originally Posted by free_to_fly

So when it asks me to find the length of the projection of OC onto the plane, is it asking me to find where OC intersects the plane and the distance between that point and the origin?

As far as I understand this question you are asked to calculate the length of c'. See attachment.