Find the angle between the pairs of lines with direction ratio proportional to: 2, 2, 1 and 4, 1, 8

Explanation

We know that, In the Cartesian or symmetrical form of equation the angle between two lines can be found by dot product equation and in this equation we will use the direction ratios which are in the denominator of the equation. `frac{z-x_{1}}{a}=frac{y-y_{1}}{b}=frac{z-z_{1}}{c}` in this equation a, b, c are the direction ratios of this equation.

So, Equations of the given lines are,


`frac{x-x_{1}}{a_{1}}=frac{y-y_{1}}{b_{1}}`
`=frac{z-z_{1}}{c_{1}}` and ` frac{x-x_{2}}{a_{2}}`
`=frac{y-y_{2}}{b_{2}}=frac{z-z_{2}}{c_{2}}`


It is given with the direction ratios and we have to find the angle between the lines having these direction ratios.


`a_{1}=2, b_{1}=2, c_{1}=1 ; a_{2}=4, b_{2}=1, c_{2}=8`

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