# arrow_back Prove that, In $R^{3}$ the distance $D$ between the point $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$ and the plane $a x+b y+c z+d=0$ is

10 views
Prove that, In $R^{3}$ the distance $D$ between the point $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$ and the plane $a x+b y+c z+d=0$ is $$D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$

Proof

Let $Q\left(x_{1}, y_{1}, z_{1}\right)$ be any point in the plane, and let $\mathbf{n}=(a, b, c)$ be a normal vector to the plane that is positioned with its initial point at $Q .$ It is now evident that the distance $D$ between $P_{0}$ and the plane is simply the length (or norm) of the orthogonal projection of the vector $\overrightarrow{Q P}_{0}$ on $\mathbf{n}$,  is
$$D=\left\|\operatorname{proj}_{\mathbf{n}} \overrightarrow{Q P_{0}}\right\|=\frac{\left|\overrightarrow{Q P}_{0} \cdot \mathbf{n}\right|}{\|\mathbf{n}\|}$$
But
\begin{aligned} &\overrightarrow{Q P_{0}}=\left(x_{0}-x_{1}, y_{0}-y_{1}, z_{0}-z_{1}\right) \\ &\overrightarrow{Q P_{0}} \cdot \mathbf{n}=a\left(x_{0}-x_{1}\right)+b\left(y_{0}-y_{1}\right)+c\left(z_{0}-z_{1}\right) \\ &\|\mathbf{n}\|=\sqrt{a^{2}+b^{2}+c^{2}} \end{aligned}
Thus
$$D=\frac{\left|a\left(x_{0}-x_{1}\right)+b\left(y_{0}-y_{1}\right)+c\left(z_{0}-z_{1}\right)\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$
Since the point $Q\left(x_{1}, y_{1}, z_{1}\right)$ lies in the given plane, its coordinates satisfy the equation of that plane; thus
$$a x_{1}+b y_{1}+c z_{1}+d=0$$
or
$$d=-a x_{1}-b y_{1}-c z_{1}$$

by Platinum
(106,844 points)

## Related questions

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
What is the distance $D$ between the point $P_{0}\left(x_{0}, y_{0},z_{0}\right)$ and the line $a x+b y+c z+d=0$ in $R^{3} ?$
What is the distance $D$ between the point $P_{0}\left(x_{0}, y_{0},z_{0}\right)$ and the line $a x+b y+c z+d=0$ in $R^{3} ?$What is the distance $D$ between the point $P_{0}\left(x_{0}, y_{0},z_{0}\right)$ and the line $a x+b y+c z+d=0$ in $R^{3} ?$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
What is the distance $D$ between the point $P_{0}\left(x_{0}, y_{0}\right)$ and the line $a x+b y+c=0$ in $R^2$?
What is the distance $D$ between the point $P_{0}\left(x_{0}, y_{0}\right)$ and the line $a x+b y+c=0$ in $R^2$?What is the distance $D$ between the point $P_{0}\left(x_{0}, y_{0}\right)$ and the line $a x+b y+c=0$ in $R^2$? ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Find the distance $D$ between the point $(1,-4,-3)$ and the plane $2 x-3 y+6 z=-1$.
Find the distance $D$ between the point $(1,-4,-3)$ and the plane $2 x-3 y+6 z=-1$.Find the distance $D$ between the point $(1,-4,-3)$ and the plane $2 x-3 y+6 z=-1$. ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
The distance from the point $(1,1,1)$ to the plane $2 x-10 y+11 z-4=0$ is equal to
The distance from the point $(1,1,1)$ to the plane $2 x-10 y+11 z-4=0$ is equal toThe distance from the point $(1,1,1)$ to the plane $2 x-10 y+11 z-4=0$ is equal to &nbsp; A) $\dfrac{1}{3}$ B) 3 C) $\dfrac{1}{15}$ D) 5 E) no ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Do the points $A(1,1,1), B(-2,0,3)$, and $C(-3,-1,1)$ form the vertices of a right triangle? Explain.
Do the points $A(1,1,1), B(-2,0,3)$, and $C(-3,-1,1)$ form the vertices of a right triangle? Explain.Do the points $A(1,1,1), B(-2,0,3)$, and $C(-3,-1,1)$ form the vertices of a right triangle? Explain. ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
The distance from the point $(-1,1,1)$ to the plane
The distance from the point $(-1,1,1)$ to the planeThe distance from the point $(-1,1,1)$ to the plane $$2 x-10 y+11 z-4=0$$ is equal to &nbsp; A) 3 B) $\frac{1}{3}$ C) 5 D) $\frac{1}{5}$ E) no ...
Determine whether planes $2 x+y+z-1=0,-x+3 y-2 z-3=0$ and $3 x-y=-1$ intersect. If yes, find the intersection.
Determine whether planes $2 x+y+z-1=0,-x+3 y-2 z-3=0$ and $3 x-y=-1$ intersect. If yes, find the intersection.Determine whether planes $2 x+y+z-1=0,-x+3 y-2 z-3=0$ and $3 x-y=-1$ intersect. If yes, find the intersection. ...