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Equation of plane passing through point vector $3\hat{i}+2\hat{j}+\hat{k}$ and parallel to vectors $2\hat{i}+\hat{j}$ and $\hat{i}+\hat{j}+\hat{k}$ is

Option 1)
$x-2y+z=0$

Option 2)
$x+2y+z=0$

Option 3)
$x+2y-z=0$

Option 4)
$x-2y-z=0$

Answers (1)

best_answer

As we have learned

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

Let the plane passes through V₀(a) and parallel to vectors U and V, then the plane is given by

$\left [ r\: U\: V \right ]= \left [a\: U\: V\right ]$

can also be written as $r= a+\lambda U+\mu V$

- wherein

$\vec{n}= \vec{u}\times \vec{v}$

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

$\left ( \vec{r} -\vec{a}\right )\cdot \vec{u}\times \vec{v}= 0$

Here, $\vec{a}=3\hat{i}+2\hat{j}+\hat{k}, \vec{u}=2\hat{i}+\hat{j}, \vec{v}=\hat{i}+\hat{j}+\hat{k}$

$\therefore$ equation will be $(\vec{r}-\vec{a})\cdot(\vec{u}\times\vec{v})=0$

$\Rightarrow$ $\begin{vmatrix} x-x_1 &y-y_1 &z-z_1 \\ 2 & 1 &0 \\ 1 &1 &1 \end{vmatrix}=0$

$\Rightarrow\begin{vmatrix} x-3 &y-2 &z-1 \\ 2& 1 &0 \\ 1& 1 &1 \end{vmatrix}=0$ 4

On solving, we get x-2y+z=0

Option 1)

$x-2y+z=0$

Option 2)

$x+2y+z=0$

Option 3)

$x+2y-z=0$

Option 4)

$x-2y-z=0$

Posted by

Himanshu

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