Question Type 4: Calculating the angle between a line and a plane Exercises

Find the acute angle between the line $\mathbf{r}=t(2,-1,3)$ and the plane $x+2y+2z=5$, giving the result in degrees to two decimal places.

Calculate the acute angle between the line $\mathbf{r} = t \begin{pmatrix} 3 \\ 5 \\ -2 \end{pmatrix}$ and the plane $2x + 3y - 6z = 4$, to two decimal places in degrees.

Find the acute angle between the line $\boldsymbol{r} = t \begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}$ and the plane $3x + 4y + 12z = 8$, giving your answer in degrees to two decimal places.

Calculate the acute angle between the line $\mathbf{r} = t \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix}$ and the plane $-x+2y+2z=3$, to two decimal places in degrees.

Calculate the acute angle between the line given by $\mathbf{r}=(3,2,1)+t(1,4,1)$ and the plane $2x+6y+z=1$, giving your answer in degrees to two decimal places.

Determine the acute angle, in degrees, between the line $r=t(4,4,1)$ and the plane $7x-2y+3z=9$, giving the result to two decimal places.

Determine the acute angle between the line $\mathbf{r}=t(6,-3,2)$ and the plane $x-4y+8z=5$, giving your answer in degrees to two decimal places.

Find the angle between the line $\mathbf{r} = t \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$ and the plane $5x + 5y + z = 2$, giving your answer in degrees to two decimal places.

Determine the acute angle between the line $\mathbf{r} = t \begin{pmatrix} 0 \\ 3 \\ 4 \end{pmatrix}$ and the plane $3x - 4y + 12z = 7$, in degrees to two decimal places.

Calculate the acute angle between the line $\mathbf{r}=t(1,-1,0)$ and the plane $y+z=1$, in degrees.

Find the angle between the line $\mathbf{r}=t(1,1,1)$ and the plane $x+y+z=0$, expressing your answer in degrees.

Determine the acute angle between the line $\mathbf{r}=t(2,2,3)$ and the plane $4x-4y+z=10$, giving the result in degrees to two decimal places.