Find the angle between the line

and the plane
Find the angle between the line

and the plane

The given line is

$⇒\stackrel{\to }{\mathrm{r}}=\left(2\stackrel{\to }{\mathrm{i}}–\stackrel{\to }{\mathrm{j}}+\stackrel{\to }{\mathrm{k}}\right)+\mathrm{t}\left(3\stackrel{\to }{\mathrm{i}}\stackrel{\to }{–\mathrm{j}}–2\stackrel{\to }{\mathrm{k}}\right)$

\Rightarrow \overrightarrow{\mathrm{r}}=(2 \overrightarrow{\mathrm{i}}-\overrightarrow{\mathrm{j}}+\overrightarrow{\mathrm{k}})+\mathrm{t}(3 \overrightarrow{\mathrm{i}} \overrightarrow{-\mathrm{j}}-2 \overrightarrow{\mathrm{k}})

the vector parallel to the given line is

$\stackrel{\to }{r}=3\stackrel{\to }{i}–\stackrel{\to }{j}–2\stackrel{\to }{k}$

\vec{r}=3 \vec{i}-\vec{j}-2 \vec{k}

The given plane is 4y+

$⇒\stackrel{\to }{r}–\left(3\stackrel{\to }{i}+4\stackrel{\to }{j}–\stackrel{\to }{k}\right)=5$

\Rightarrow \vec{r}-(3 \vec{i}+4 \vec{j}-\vec{k})=5

the normal to the given plane is

$\stackrel{\to }{\stackrel{\to }{n}}=3\stackrel{\to }{i}+4\stackrel{\to }{j}+\stackrel{\to }{k}$

\overrightarrow{\vec{n}}=3 \vec{i}+4 \vec{j}+\vec{k}

Let be the angle between the line and the plane, then