## Foot of Perpendicular is a hot topic for H2 Prelims and A Levels. It comes out almost every year.

There are two versions of Foot of Perpendicular, from point to line, and from point to plane. However, the two are highly similar, and the following article will teach how to understand and remember them.

## H2: Vectors (Foot of perpendicular)

From point (B) to Line ( $l$)

(Picture)

### Equation (I):

Where does F lie? F lies on the line  $l$.

$\overrightarrow{\mathit{OF}}=\mathbf{a}+\lambda \mathbf{m}$

### Equation (II):

Perpendicular:

$\overrightarrow{\mathit{BF}}\cdot \mathbf{m}=0$

$(\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OB}})\cdot \mathbf{m}=0$

### Final Step

Substitute Equation (I) into Equation (II) and solve for  $\lambda$.

## Example:

[CJC 2010 P1Q7iii]

Relative to the origin $O$ , the points $A$ , $B$ and $C$  have position vectors  $\left(\begin{matrix}1\\2\\1\end{matrix}\right)$ , $\left(\begin{matrix}2\\1\\3\end{matrix}\right)$ and $\left(\begin{matrix}-1\\2\\3\end{matrix}\right)$ Find the shortest distance from  $C$ to $\mathit{AB}$ . Hence or otherwise, find the area of triangle $\mathit{ABC}$ .

[Note: There is a 2nd method to this question. (cross product method)]

## Solution:

Let the foot of perpendicular from C to AB be F.

Equation (I):

$\overrightarrow{\mathit{OF}}=\overrightarrow{\mathit{OA}}+\lambda \overrightarrow{\mathit{AB}}=\left(\begin{matrix}1+\lambda \\2-\lambda \\1+2\lambda \end{matrix}\right)$

Equation (II):

$(\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OC}})\cdot \overrightarrow{\mathit{AB}}=0$

$\left(\begin{matrix}2+\lambda \\-\lambda \\-2+2\lambda \end{matrix}\right)\cdot \left(\begin{matrix}1\\-1\\2\end{matrix}\right)=0$

$\lambda =\frac{1}{3}$

$\overrightarrow{\mathit{CF}}=\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OC}}=\left(\begin{matrix}2\frac{1}{3}\\-{\frac{1}{3}}\\-1\frac{1}{3}\end{matrix}\right)$

$\left|{\overrightarrow{{\mathit{CF}}}}\right|=\sqrt{\frac{22}{3}}$

Area of  $\Delta \mathit{ABC}=\frac{1}{2}\left|{\overrightarrow{\mathit{AB}}}\right|\left|{\overrightarrow{\mathit{CF}}}\right|=\sqrt{11}$

For the next part, please read our article on Foot of Perpendicular (from point to plane).

## H2 Maths Tuition

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