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SHORTEST DISTANCE (POINT TO PLANE) 
$=\left| \overrightarrow{AP}\cdot \mathbf{\hat{n}} \right|$ 
Hence, Shortest Distance to Origin ${\left(\overrightarrow{OP} = \mathbf{O}\right) \Rightarrow \left| \mathbf{a} \cdot \hat{\mathbf{d}} \right|}$
FINDING THE COORDINATES OF FOOT OF PERPENDICULAR, $\overrightarrow{ON}$
1.	Find line equation ${PN:\ \overrightarrow{PN} = \mathbf{p} + \lambda \mathbf{n}}$ for some ${\lambda \in \mathbb{R}}$. 
2.	Solve $\lambda $ by finding the intersection point of line $PN$ and the plane. 
3.	Substitute ${\lambda}$ into ${\overrightarrow{PN} = \mathbf{p} + \lambda n\mathbf{P}}$ to find ${\overrightarrow{ON}}$. 
$A$ is any point on the plane, it can be obtained by substituting values which satisfy the equation of plane. 
For example, $\prod :\mathbf{r}\cdot \left( \begin{matrix}
 2 \\
 -1 \\
 3 \\
\end{matrix} \right)=4\Rightarrow $$A$ can be $\left( \begin{matrix}
 2 \\
 0 \\
 0 \\
\end{matrix} \right)$, $\left( \begin{matrix}
 0 \\
 -4 \\
 0 \\
\end{matrix} \right)$ or $\left( \begin{matrix}
 1 \\
 -2 \\
 0 \\
\end{matrix} \right)$.

$=\left| \overrightarrow{AP}\cdot \mathbf{\hat{n}} \right|$


Hence, Shortest Distance to Origin ${\left(\overrightarrow{OP} = \mathbf{O}\right) \Rightarrow \left| \mathbf{a} \cdot \hat{\mathbf{d}} \right|}$

FINDING THE COORDINATES OF FOOT OF PERPENDICULAR, $\overrightarrow{ON}$

1.	Find line equation ${PN:\ \overrightarrow{PN} = \mathbf{p} + \lambda \mathbf{n}}$ for some ${\lambda \in \mathbb{R}}$.


2.	Solve $\lambda $ by finding the intersection point of line $PN$ and the plane. 


3.	Substitute ${\lambda}$ into ${\overrightarrow{PN} = \mathbf{p} + \lambda n\mathbf{P}}$ to find ${\overrightarrow{ON}}$.

$A$ is **any** point on the plane, it can be obtained by substituting values which satisfy the equation of plane. 


For example, $\prod :\mathbf{r}\cdot \left( \begin{matrix}
   2  \\
   -1  \\
   3  \\
\end{matrix} \right)=4\Rightarrow $$A$ can be $\left( \begin{matrix}
   2  \\
   0  \\
   0  \\
\end{matrix} \right)$, $\left( \begin{matrix}
   0  \\
   -4  \\
   0  \\
\end{matrix} \right)$ or $\left( \begin{matrix}
   1  \\
   -2  \\
   0  \\
\end{matrix} \right)$. 


Relationships Between a Point and a Plane

Vectors

International Baccalaureate Higher Level Mathematics

Relationships Between a Point and a Plane

Relationships Between a Point and a Plane

Solution:

Handwritten & Typed Solutions