Question

Reduce the equation \[\overrightarrow{r}.\left( \overrightarrow{i}-2\overrightarrow{j}+2\overrightarrow{k} \right)\text{ }-\text{ }6\text{ }=\text{ }0\] to normal form and hence find the length of perpendicular from the origin to the plane.

Answer 1

Hint: Convert the given equation in the form of \[\overrightarrow{r}.\overrightarrow{n}=\text{p}\] and then use the formula of given by \[\overrightarrow{r}.\dfrac{\overrightarrow{n}}{\left| \overrightarrow{n} \right|}=\dfrac{\text{p}}{\left| \overrightarrow{n} \right|}\] to get the normal form of the given equation. Also, the value of \[\dfrac{\text{p}}{\left| \overrightarrow{n} \right|}\] is the perpendicular distance from the origin.

Complete step-by-step answer:
To solve the above problem we will write the given equation first, therefore,
\[\overrightarrow{r}.\left( \overrightarrow{i}-2\overrightarrow{j}+2\overrightarrow{k} \right)\text{ }-\text{ }6\text{ }=\text{ }0\]
If we shift -6 on the right hand side of the equation we will get,
\[\therefore \overrightarrow{r}.\left( \overrightarrow{i}-2\overrightarrow{j}+2\overrightarrow{k} \right)\text{ }=\text{ 6}\]
If we compare the above equation with \[\overrightarrow{r}.\overrightarrow{n}=\text{p}\] we will get,
\[\overrightarrow{n}=\left( \overrightarrow{i}-2\overrightarrow{j}+2\overrightarrow{k} \right)\] and p = 6 …………………………………………………………………….. (1)
Now we have to find out \[\left| \overrightarrow{n} \right|\] to reduce the equation in to normal form and for that we should know the formula given below,
Formula:
If \[\overrightarrow{n}=a\overrightarrow{i}+b\overrightarrow{j}+c\overrightarrow{k}\] then \[\left| \overrightarrow{n} \right|\] is given by \[\left| \overrightarrow{n} \right|=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]
By using the formula and the values of equation (1) we will find the value of \[\left| \overrightarrow{n} \right|\] for given equation as follows,
\[\therefore \left| \overrightarrow{n} \right|=\sqrt{{{1}^{2}}+{{\left( -2 \right)}^{2}}+{{\left( 2 \right)}^{2}}}\]
\[\therefore \left| \overrightarrow{n} \right|=\sqrt{1+4+4}\]
\[\therefore \left| \overrightarrow{n} \right|=\sqrt{9}\]
As we know that the square root of 9 is 3 therefore above equation will become,
\[\therefore \left| \overrightarrow{n} \right|=3\] …………………………………………………. (2)
Now, to reduce the equation in the normal form we should know the formula given below,
Formula:
The normal form of the equation \[\overrightarrow{r}.\overrightarrow{n}=\text{p}\] is given by \[\overrightarrow{r}.\dfrac{\overrightarrow{n}}{\left| \overrightarrow{n} \right|}=\dfrac{\text{p}}{\left| \overrightarrow{n} \right|}\].
If we put the value of equation (1) and equation (2) in the above formula we will get,
\[\therefore \overrightarrow{r}.\dfrac{\left( \overrightarrow{i}-2\overrightarrow{j}+2\overrightarrow{k} \right)}{3}=\dfrac{6}{3}\]
If we simplify the above equation we will get,
\[\therefore \overrightarrow{r}.\left( \dfrac{\overrightarrow{i}}{3}-\dfrac{2\overrightarrow{j}}{3}+\dfrac{2\overrightarrow{k}}{3} \right)=2\] ……………………………………………………… (4)
Therefore the normal form of the equation is \[\overrightarrow{r}.\left( \overrightarrow{i}-2\overrightarrow{j}+2\overrightarrow{k} \right)\text{ }-\text{ }6\text{ }=\text{ }0\] is \[\overrightarrow{r}.\left( \dfrac{1}{3}\overrightarrow{i}-\dfrac{2}{3}\overrightarrow{j}+\dfrac{2}{3}\overrightarrow{k} \right)=2\].
As we know that \[\dfrac{\overrightarrow{n}}{\left| \overrightarrow{n} \right|}=\overset{\hat{\ }}{\mathop{n}}\,\] and therefore the normal form of the equation can also be written as \[\overrightarrow{r}.\overset{\hat{\ }}{\mathop{n}}\,=d\] where ‘d’ is the distance of the plane from the origin which is the required perpendicular distance from the origin.
As the equation (4) is of the form \[\overrightarrow{r}.\overset{\hat{\ }}{\mathop{n}}\,=d\] therefore,
d = 2
Therefore the length of perpendicular from the origin is equal to 2.

Note: Always remember that the equation \[\overrightarrow{r}.\overrightarrow{n}=\text{p}\] and \[\overrightarrow{r}.\overset{\hat{\ }}{\mathop{n}}\,=d\] are not same as many students write the value of p in place of d which results in incorrect answer.