Question

Show that the given lines intersect and find the acute angle between them.

$$\begin{gathered} \mathbf{r}\mathbf{=}\mathbf{(}\mathbf{5}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{+}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{1}\mathbf{)}{\mathbf{t}}_{\mathbf{1}}\mathbf{and} \\ \mathbf{r}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{,}\mathbf{-}\mathbf{4}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{+}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{)}{\mathbf{t}}_{\mathbf{2}} \end{gathered}$$

Short answer

The Acute angle between the lines is$\theta =36.8$

Step-by-step solution

Given Information:

Equation of two lines are as follows,

$$\begin{gathered} r=(5,-2,0)+(1,-1-1)t_1\mathrm{and} \\ r=(4,-4,-1)+(0,3,2)t_2 \end{gathered}$$

Concept used

Equating both the line equations and substituting the value for the variable gives the point of intersection. Finding the angle between the two given lines using formula for gives the acute angle between the lines.

Equation representing the point of intersection

$(5,-2,0)+(1,-1-1)t_1=(4,-4,-1)+(0,3,2)t_2$

This gives 3 equations

$$\begin{gathered} 5+t_1=4\to t_1=-1 \\ -2-t_1=4+3t_2 \\ -t_1=-1+2t_2 \end{gathered}$$

Substitute the second equation in the third one.

$-(-1)=-1+2t_1\to -2t_2=-2\to t_2$

So, the point of intersection is given by substituting $t_1=-1\mathrm{or}{t}_2=1$And both gives $r=4i-j+1k.$

To find a vector parallel to each line.

The two vectors are as follows

$$\begin{gathered} A=i-j-k \\ B=3j+2k \end{gathered}$$

Let the angle between the A vectors B or$\theta$

$$\begin{gathered} A\cdot B=AB \\ \cos \theta \cos \theta =(A\cdot B)/AB \end{gathered}$$

$$\begin{gathered} A=\left|A\right|=\sqrt{(1{)}^2+(-1{)}^2+(-1{)}^2}=\sqrt{3} \\ B=\left|B\right|=\sqrt{(0{)}^2+(3{)}^2+(2{)}^2}=\sqrt{13} \\ A\times B=\left(1\right)\left(0\right)+(-1)\left(3\right)+(-1)\left(2\right)=-3-2=-5 \\ \cos \theta =\frac{A\times B}{AB}=\frac{-5}{\sqrt{13\times 3}} \\ \theta =co{s}^{-1}\frac{5}{\sqrt{39}}=143.2 \end{gathered}$$

But the acute angle is$180-\theta =180-143.2=36.8$

The angle between the lines is $\theta =36.8^0$.