Answer

Verified

446.4k+ views

**Hint:**

Firstly, we calculate the intersecting points of the lines. Calculate the cross product of the planes such as the cross product of plane 1 and plane 2 and the cross product of plane 3 and plane 4.

**Complete step by step solution:**

The given line equations are $x - y + z = 1,3x - y - z + 1 = 0$ and $x + 4y - z = 1,x + 2y + z + 1 = 0$.

Now, we take the first two plane equations and suppose that plane 1 is ${p_1} = x - y + z = 1$ and plane 2 is ${p_2} = 3x - y - z + 1 = 0$. There will be a line of intersection.

We suppose that $z = 0$ and calculate the line of intersection of planes. Substitute the value $z = 0$ in the expression ${p_1} = x - y + z = 1$.

$x - y + \left( 0 \right) = 1\\

\Rightarrow x - y = 1$ (i)

Again, we substitute the value $z = 0$ in the expression ${p_2} = 3x - y - z + 1 = 0$.

$3x - y - \left( 0 \right) + 1 = 0\\

\Rightarrow 3x - y = - 1$ (ii)

Now, subtract equation (i) from equation (ii):

$3x - y - x + y = - 1 - 1\\

\Rightarrow 2x = - 2\\

\Rightarrow x = - 1$

We substitute the value $x = - 1$ in the expression $x - y = 1$.

$- 1 - y = 1\\

\Rightarrow - y = 2\\

\Rightarrow y = - 2$

Hence, the line of intersection is ${L_1} = \left( { - 1, - 2,0} \right)$.

Let us suppose the direction of the line is $\overline a $. The normal of plane 1 is ${n_1}$ and normal of plane 2 is ${n_2}$.

The cross product of the normal plane is known as the direction of line.

$\overline a = \overline {{n_1}} \times \overline {{n_2}} \\

= \left| {\begin{array}{*{20}{c}}

{\widehat i}&{\widehat j}&{\widehat k}\\

1&{ - 1}&1\\

3&{ - 1}&{ - 1}

\end{array}} \right|\\

= \widehat i\left( {1 + 1} \right) - \widehat j\left( { - 1 - 3} \right) + \widehat k\left( { - 1 + 3} \right)\\

= 2\widehat i + 4\widehat j + 2\widehat k$

Hence, the equation of ${L_1} = \left( { - 1, - 2,0} \right)$ is ${L_1} = - \widehat i - 2\widehat j + 0\widehat k + \lambda \left( {2\widehat i + 4\widehat j + 2\widehat k} \right)$.

Take the last two plane equations and suppose plane 3 is ${p_3} = x + 4y - z = 1$ and plane 4 is ${p_4} = x + 2y + z + 1 = 0$. There will be a line of intersection.

Let us suppose $z = 0$ and calculate the line of intersection of planes. Substitute $z = 0$ in the expression ${p_3} = x + 4y - z = 1$.

$x + 4y - 0 = 1\\

\Rightarrow x + 4y = 1$ (iii)

Again, Substitute $z = 0$ in the expression ${p_4} = x + 2y + z + 1 = 0$.

$x + 2y + 0 + 1 = 0\\

\Rightarrow x + 2y = - 1$ (iv)

Now, subtract equation (iii) from equation (iv):

$x + 2y - x - 4y = - 1 - 1\\

\Rightarrow - 2y = - 2\\

\Rightarrow y = 1$

Substitute $y = 1$ in the expression $x + 4y = 1$.

$\Rightarrow x + 4\left( 1 \right) = 1\\

\Rightarrow x = 1 - 4\\

\Rightarrow x = - 3$

Hence, the line of intersection is ${L_2} = \left( { - 3,1,0} \right)$.

Let us suppose the direction of the line is $\overline b $. The normal of plane 3 is ${n_3}$ and normal of plane 4 is ${n_4}$.

The cross product of the normal plane is known as the direction of line.

$

\overline b = \overline {{n_3}} \times \overline {{n_4}} \\

= \left| {\begin{array}{*{20}{l}}

{\widehat i}&{\widehat j}&{\widehat k}\\

1&4&{ - 1}\\

1&2&1

\end{array}} \right|\\

= \widehat i\left( {4 + 2} \right) - \widehat j\left( {1 + 1} \right) + \widehat k\left( {2 - 4} \right)\\

= 6\widehat i - 2\widehat j - 2\widehat k$

Hence, the equation of ${L_2} = \left( { - 3,1,0} \right)$ is ${L_2} = - 3\widehat i + \widehat j + 0\widehat k + \mu \left( {6\widehat i - 2\widehat j - 2\widehat k} \right)$.

Here the direction vectors $\overline a $ is not parallel to the direction vector $\overline b $. Hence, the line is not parallel to each other.

Now, check the intersection in between the planes:

The coordinate of x in line one and line two is $ - 1 + 2\lambda = - 3 + 6\mu $.

Similarly, the coordinate of y in line one and line two is $ - 2 + 4\lambda = 1 - 2\mu $ and the coordinate of z in line one and line two is $2\lambda = - 2\mu $ which means $\lambda = - \mu $.

Substitute $\lambda = - \mu $ in the expression $ - 1 + 2\lambda = - 3 + 6\mu $.

$ - 1 + 2\left( { - \mu } \right) = - 3 + 6\mu \\

\Rightarrow - 1 - 2\mu = - 3 + 6\mu \\

\Rightarrow 8\mu = - 1 + 3\\

\Rightarrow \mu = \dfrac{1}{4}$

Substitute $\lambda = - \mu $ in the expression $ - 2 + 4\lambda = 1 - 2\mu $.

$- 2 + 4\left( { - \mu } \right) = 1 - 2\mu \\

\Rightarrow - 2 - 4\mu = 1 - 2\mu \\

\Rightarrow - 2\mu = 1 + 3\\

\Rightarrow \mu = - 2$

Hence, the intersecting point does not exist because the value of $\mu $ is different from the same value of $\lambda $. So, no real point of intersection.

Therefore, the line does not intersect with each other and is not parallel to each other; it means the lines are skew lines.

Now, check perpendicular and non-perpendicular of the direction of the line. That is:

**Hence, the line is perpendicular to each other. Therefore, the option (C) is the correct option.**

**Note:**

Here you should know what intersection points, skew points, parallel lines, and perpendicular lines. Parallel is basically the lines never intersect to each other. The perpendicular line means the lines which lie on right angles means ${90^ \circ}$. Skew lines are the lines which have no intersections and are not parallel.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

A rainbow has circular shape because A The earth is class 11 physics CBSE

Which are the Top 10 Largest Countries of the World?

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

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell