tough matrix problem!! can you help me solve it?

[wildwildwest568]

Given: The point A(x0, y0, z0) lies outside a plane ? with equation ax+by+cz+d=0. The point P(x, y, z) lies on the plane ?. AB is the distance between the point A and the plane ?.

Derive: The equation: (AB)= | ax0+by0+cz0+d | / v(a^2+b^2+c^2)

(the bars are an absolute value and the v is a square root sign)

ANY input (advice, strategies, comments, etc) is much appreciated

THANKS!!

 

[soroban]

Hello, wildwildwest568!

Here's a game plan for you . . .

Given: The point \(\displaystyle A(x_o, y_o, z_o)\) lies outside a plane \(\displaystyle \pi\) with equation \(\displaystyle ax\,+\,by\,+\,cz\,+\,d\:=\:0\).

The point \(\displaystyle P(x, y, z)\) lies on the plane \(\displaystyle \pi\).

\(\displaystyle d(AB)\) is the distance between the point \(\displaystyle A\) and the plane \(\displaystyle \pi\).

Derive the equation: \(\displaystyle \L\; d(AB)\;=\;\frac{\left|ax_o\,+\,by_o\,+\,cz_o\,+\,d\right|}{\sqrt{a^2\,+\,b^2\,+\,c^2}}\)

We want line \(\displaystyle L\) through point \(\displaystyle A\) and perpendicular to the plane.

The plane has normal vector: \(\displaystyle \,\vec{n}\:=\:\langle a,b,c\rangle\)

Hence, line \(\displaystyle L\) has parametric equations: \(\displaystyle \:\begin{array}{cc}x\:=\:x_o\,+\,at \\ y\:=\:y_o\,+\,bt \\ z\:=\:z_o\,+\,ct\end{array}\)

Find point \(\displaystyle B\), the intersection of line \(\displaystyle L\) and plane \(\displaystyle \pi\).
\(\displaystyle \;\;\)(Substitute the parametric equation of \(\displaystyle L\) into the equation of the plane and solve for \(\displaystyle t\).)

Then use the Distance Formula on points \(\displaystyle A\) and \(\displaystyle B\).

 

[wildwildwest568]

thanks soroban!!

that helped me get started, but if anyone else has other input, that would be great...

 

[wildwildwest568]

oh, and how did you get it all formatted right?

 

[stapel]

how did you get it all formatted right?

The tutor used LaTeX. To learn the basic commands, try the links in the "Forum Help" pull-down menu at the very top of the page.

Eliz.

 

[wildwildwest568]

ok thanks.

does anyone else have ideas about solving?

 

[stapel]

does anyone else have ideas about solving?

Where are you stuck?

Please reply showing all of the steps you have tried.

Thank you.

Eliz.