# Write the sum of intercepts cut off by the place \[\overrightarrow{r}.\left( 2\widehat{i}+\widehat{j}-\widehat{k} \right)-5=0\] on the three axes.

Last updated date: 19th Mar 2023

•

Total views: 303.6k

•

Views today: 3.85k

Answer

Verified

303.6k+ 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

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

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

Name the Largest and the Smallest Cell in the Human Body ?