Answer

Verified

472.8k+ views

Hint: Here, first off substitute \[r=x\widehat{i}+y\widehat{j}+z\widehat{k}\] to find the equation of plane in ax + by + cz + d = 0 form. Then we know that plane in intercept form is \[\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\] where a, b and c are intercepts on x, y and z axes. So, convert the given plane into the plane in the intercept form to get the sum of the intercepts.

Complete step by step solution:

Here we have to find the sum of the intercepts cut off by the plane \[\overrightarrow{r}.\left( 2\widehat{i}+\widehat{j}-\widehat{k} \right)-5=0\] on three axes.

Let us first consider the equation of the plane given in the question.

\[P=\overrightarrow{r}.\left( 2\widehat{i}+\widehat{j}-\widehat{k} \right)-5=0\]

We know that \[\overrightarrow{r}=x\widehat{i}+y\widehat{j}+z\widehat{k}\]. By substituting the value of \[\overrightarrow{r}\] in the above equation, we get,

\[P:\left( x\widehat{i}+y\widehat{j}+z\widehat{k} \right).\left( 2\widehat{i}+\widehat{j}-\widehat{k} \right)-5=0\]

We know that \[\left( a\widehat{i}+b\widehat{j}+c\widehat{k} \right).\left( m\widehat{i}+n\widehat{j}+q\widehat{k} \right)=am+bn+cq\]

By using this, we get,

\[P:2x+y-z-5=0\]

By dividing the whole equation by 5, we get,

\[P:\dfrac{2x}{5}+\dfrac{y}{5}-\dfrac{z}{5}=1\]

We can also write the above equation as,

\[P:\dfrac{x}{\dfrac{5}{2}}+\dfrac{y}{5}+\dfrac{z}{\left( -5 \right)}=1....\left( i \right)\]

We know that the equation of a plane in intercept form is given by \[\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\] where a, b and c are the intercepts cut off by plane in x, y and z axes respectively.

By comparing equation (i) with equation of plane in the intercept form that is \[\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\]

We get,

\[\begin{align}

& a=\dfrac{5}{2} \\

& b=5 \\

& c=-5 \\

\end{align}\]

Hence, we get,

Intercept cut off by the place on the x axis \[=\dfrac{5}{2}\]

Intercept cut off by the plane on y-axis = 5

Intercept cut off by the place on z-axis = -5

Hence, we get the sum of the intercept cut off by plane on all axes

\[=\dfrac{5}{2}+5-5\]

\[=\dfrac{5}{2}=2.5\]

Therefore, we have formed the sum of the intercepts as \[\dfrac{5}{2}=2.5\]

Note: Students must note that the negative intercept signifies that the intercept cut off by the given plane is in the negative axis. For example, if we get ‘-a’ as an intercept on the x-axis, that means the intercept cut off by plane is ‘a’ on the negative x-axis or on the left side of the origin. Also, students are advised to always first substitute \[\overrightarrow{r}=x\widehat{i}+y\widehat{j}+z\widehat{k}\] and then solve the questions related to the plane to get the answers easily.

Complete step by step solution:

Here we have to find the sum of the intercepts cut off by the plane \[\overrightarrow{r}.\left( 2\widehat{i}+\widehat{j}-\widehat{k} \right)-5=0\] on three axes.

Let us first consider the equation of the plane given in the question.

\[P=\overrightarrow{r}.\left( 2\widehat{i}+\widehat{j}-\widehat{k} \right)-5=0\]

We know that \[\overrightarrow{r}=x\widehat{i}+y\widehat{j}+z\widehat{k}\]. By substituting the value of \[\overrightarrow{r}\] in the above equation, we get,

\[P:\left( x\widehat{i}+y\widehat{j}+z\widehat{k} \right).\left( 2\widehat{i}+\widehat{j}-\widehat{k} \right)-5=0\]

We know that \[\left( a\widehat{i}+b\widehat{j}+c\widehat{k} \right).\left( m\widehat{i}+n\widehat{j}+q\widehat{k} \right)=am+bn+cq\]

By using this, we get,

\[P:2x+y-z-5=0\]

By dividing the whole equation by 5, we get,

\[P:\dfrac{2x}{5}+\dfrac{y}{5}-\dfrac{z}{5}=1\]

We can also write the above equation as,

\[P:\dfrac{x}{\dfrac{5}{2}}+\dfrac{y}{5}+\dfrac{z}{\left( -5 \right)}=1....\left( i \right)\]

We know that the equation of a plane in intercept form is given by \[\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\] where a, b and c are the intercepts cut off by plane in x, y and z axes respectively.

By comparing equation (i) with equation of plane in the intercept form that is \[\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\]

We get,

\[\begin{align}

& a=\dfrac{5}{2} \\

& b=5 \\

& c=-5 \\

\end{align}\]

Hence, we get,

Intercept cut off by the place on the x axis \[=\dfrac{5}{2}\]

Intercept cut off by the plane on y-axis = 5

Intercept cut off by the place on z-axis = -5

Hence, we get the sum of the intercept cut off by plane on all axes

\[=\dfrac{5}{2}+5-5\]

\[=\dfrac{5}{2}=2.5\]

Therefore, we have formed the sum of the intercepts as \[\dfrac{5}{2}=2.5\]

Note: Students must note that the negative intercept signifies that the intercept cut off by the given plane is in the negative axis. For example, if we get ‘-a’ as an intercept on the x-axis, that means the intercept cut off by plane is ‘a’ on the negative x-axis or on the left side of the origin. Also, students are advised to always first substitute \[\overrightarrow{r}=x\widehat{i}+y\widehat{j}+z\widehat{k}\] and then solve the questions related to the plane to get the answers easily.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE