# The plane containing the line  and parallel to the line     passes through the point : Option 1) (1, -2, 5) Option 2) (1, 0, 5) Option 3) (0, 3, -5) Option 4) (-1, -3, 0)

As we learnt in

Plane passing through a point and parallel to two given vectors (Cartesian form) -

Let the plane passes through $A(x_{1},y_{1},z_{1})$and parallel to vectors having DR's $(a_{1},b_{1},c_{1})\: and \: (a_{2},b_{2},c_{2})$, then the plane is given by

$\begin{vmatrix} x-x_{1} & y-y_{1} &z-z_{1} \\ a_{1} & b_{1} &c_{1} \\ a_{2}& b_{2} & c_{2} \end{vmatrix}=0$

- wherein

$\vec{n}=\begin{vmatrix} \hat{i} & \hat{j} &\hat{k} \\ a_{1} & b_{1} &c_{1} \\ a_{2}& b_{2} & c_{2} \end{vmatrix}$

$\left ( \vec{r}-\vec{a} \right )\cdot \vec{n}= 0$

Line is $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$

The normal vector of plane is

$\begin{vmatrix} \hat{i} & \hat{j}& \hat{k}\\ 1& 2 &3 \\ 1 &1 &4 \end{vmatrix}=5\hat{i}-\hat{j}-\hat{k}$

So equation is 5x-y-z = 5-2-3=0

5x-y-z=0

It passes through (1,0,5)

Option 1)

(1, -2, 5)

This option is incorrect

Option 2)

(1, 0, 5)

This option is correct

Option 3)

(0, 3, -5)

This option is incorrect

Option 4)

(-1, -3, 0)

This option is incorrect

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