Show that the plane whose vector equation is r ( i +2 j - k )=3 contains the line whose vector equation is r = i + j + (2 i + j +4 k ).

The line r =( i + j +0 k )+ (2 i + j +4 k ) ...(i) Passes through a point whose posotion vector is a = i + j +0 k and is parallel to the vector b =2 i + j +4 k If the plane r ( i +2 j - k )=3 contains the given line, then It should passes through the point i + j +0 k . It should be parallel to the line. Now, the plane passes through the point i + j +0 k . So, the plane vector to the given plane is n = i +2 j - k . We observe that b n =(2 i + j +4 k ) ( i +2 j - k ) =2+2-4=0 Therefore, the plane is parallel to the line. Hence, the given plane contains the given line.

Show that the plane whose vector equation is r ( i +2 j - k )=3 c…

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Question

Show that the plane whose vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=3$ contains the line whose vector equation is $\vec{\text{r}}=\hat{\text{i}}+\hat{\text{j}}+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}).$

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Answer

The line $\vec{\text{r}}=(\hat{\text{i}}+\hat{\text{j}}+0\hat{\text{k}})+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}})\ ...(\text{i})$
Passes through a point whose posotion vector is $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+0\hat{\text{k}}$ and is parallel to the vector $\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}$
If the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=3$ contains the given line, then
It should passes through the point $\hat{\text{i}}+\hat{\text{j}}+0\hat{\text{k}}.$
It should be parallel to the line.
Now, the plane passes through the point $\hat{\text{i}}+\hat{\text{j}}+0\hat{\text{k}}.$
So, the plane vector to the given plane is $\vec{\text{n}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}.$
We observe that
$\vec{\text{b}}\cdot\vec{\text{n}}=(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}})\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})$
$=2+2-4=0$
Therefore, the plane is parallel to the line.
Hence, the given plane contains the given line.