Fundraise on MathsGee
First time here? Checkout the FAQs!
x

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

1 like 0 dislike
251 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 $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}} $$
in Mathematics by Platinum (130,996 points) | 251 views

1 Answer

0 like 0 dislike
Best answer

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 (130,996 points)

Related questions

0 like 0 dislike
0 answers
0 like 0 dislike
1 answer

Join the MathsGee Learning Club where you get study and financial support for success from our community. CONNECT - LEARN - FUNDRAISE


On the MathsGee Learning Club, you can:


1. Ask questions


2. Answer questions


3. Vote on Questions and Answers


4. Start a Fundraiser


5. Tip your favourite community member(s)


6. Create Live Video Tutorials (Paid/Free)


7. Join Live Video Tutorials (Paid/Free)


8. Earn points for participating



Posting on the MathsGee Learning Club


1. Remember the human


2. Behave like you would in real life


3. Look for the original source of content


4. Search for duplicates before posting


5. Read the community's rules




CLUB RULES


1. Answers to questions will be posted immediately after moderation


2. Questions will be queued for posting immediately after moderation


3. Depending on how many posts we receive, you could be waiting up to 24 hours for your post to appear. But, please be patient as posts will appear after they pass our moderation.


MathsGee Android Q&A