H2 Maths Tuition: Foot of Perpendicular (from point to plane) (Part II)

This is a continuation from H2 Maths Tuition: Foot of Perpendicular (from point to line) (Part I).

Foot of Perpendicular (from point to plane)

From point (B) to Plane ( $p$ )

Equation (I):

Where does F lie?

F lies on the plane $p$ .

$\overrightarrow{\mathit{OF}}\cdot \mathbf{n}=d$

Equation (II):

Perpendicular

$\overrightarrow{\mathit{BF}}=k\mathbf{n}$

$\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OB}}=k\mathbf{n}$

$\overrightarrow{\mathit{OF}}=k\mathbf{n}+\overrightarrow{OB}$

Final Step

Substitute Equation (II) into Equation (I) and solve for k.

Example

[VJC 2010 P1Q8i]

The planes $\Pi _{1}$ and $\Pi _{2}$ have equations $\mathbf{r\cdot(i+j-k)}=6$ and $\mathbf{r\cdot(2i-4j+k)}=-12$ respectively. The point $A$ has position vector $\mathbf{{9i-7j+5k}}$ .

(i) Find the position vector of the foot of perpendicular from $A$ to $\Pi _{2}$ .

Solution

Let the foot of perpendicular be F.

Equation (I)

$\overrightarrow{\mathit{OF}}\cdot \left(\begin{matrix}2\\-4\\1\end{matrix}\right)=-12$

Equation (II)

$\overrightarrow{\mathit{OF}}=k\left(\begin{matrix}2\\-4\\1\end{matrix}\right)+\left(\begin{matrix}9\\-7\\5\end{matrix}\right)=\left(\begin{matrix}2k+9\\-4k-7\\k+5\end{matrix}\right)$

Subst. (II) into (I)

$2(2k+9)-4(-4k-7)+(k+5)=-12$

Solve for k, $k=-3$ .

$\overrightarrow{\mathit{OF}}=\left(\begin{matrix}3\\5\\2\end{matrix}\right)$

H2 Maths Tuition

If you are looking for Maths Tuition, contact Mr Wu at:

Email: mathtuition88@gmail.com

Author: mathtuition88

https://mathtuition88.com/ View all posts by mathtuition88

One thought on “H2 Maths Tuition: Foot of Perpendicular (from point to plane) (Part II)”

Pingback: Foot of perpendicular from point to plane | Singapore Maths Tuition

Leave a Reply Cancel reply

Enter your comment here...

Fill in your details below or click an icon to log in:

Email (required) (Address never made public)

Name (required)

Website

You are commenting using your WordPress.com account. ( Log Out / Change )

You are commenting using your Google account. ( Log Out / Change )

You are commenting using your Twitter account. ( Log Out / Change )

You are commenting using your Facebook account. ( Log Out / Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

<span>%d</span> bloggers like this: