Question Type 4: Finding the angle between two lines using the direction vectors Exercises

Calculate the acute angle between the lines $l_1: (2,1,4)+k(5,9,1)$ and $l_2: (1,0,1)+t(8,7,2)$ in degrees.

Express the acute angle $\theta$ between the lines $l_1: (2,1,4)+k(5,9,1)$ and $l_2: (1,0,1)+t(8,7,2)$ in radians.

Find $\tan\theta$ where $\theta$ is the acute angle between the lines $l_1: (2,1,4)+k(5,9,1)$ and $l_2: (1,0,1)+t(8,7,2)$.

Use the cross product method to find the acute angle between $l_1: (2,1,4)+k(5,9,1)$ and $l_2: (1,0,1)+t(8,7,2)$.

Let $d_3=d_1\times d_2$ where $d_1$ and $d_2$ are the direction vectors of $l_1$ and $l_2$. Find the angle between the line $l_1$ and the line with direction $d_3$.

If $d_1'=3d_1$ and $d_2'=-2d_2$ are scaled direction vectors of $l_1$ and $l_2$, show that the angle between the lines with $d_1'$ and $d_2'$ is the same as between $d_1$ and $d_2$.