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.