Question

The angle between the lines passing through the points (4, 7, 8), (2, 3, 4) and (-1, -2, 1), (1, 2, 5) is

Answer 1

Answer:

1) acute angle

2) right angle

3) obtuse angle

4) zero angle

Answer 2

Answer: The angle between the lines   θ = 180°

Step-by-step explanation:

The line passing through the points A (4, 7, 8), B (2, 3, 4) and C (-1, -2, 1), D (1, 2, 5)

• Directional Ratio of AB & CD

If    $(x_{1} y_{1} z_{1} )$$(x_{2} y_{2}z_{2} )$ are points then DR are $(x_{2}-x_{1} , y_{2} - y_{1},z_{2} -z_{1} )$

• So Directional Ratio for AB = (2-4 , 3-7 ,4-8) = (-2 , -4 , -4) compare with$(a_{1} b_{1} c_{1} )$

& for CD = (1-(-1) , 2-(-2) ,5-1) = (2 , 4 ,4) compare with $(a_{2} b_{2} c_{2} )$

• If θ is the angle between 2 lines whose D.R are $(a_{1} b_{1} c_{1} )$  $(a_{2} b_{2} c_{2} )$

Then ,

Cosθ =$\frac{a_{1}a_{2}+b_{1} b_{2} +c_{1} c_{2} }{\sqrt{a_{1} ^{2}+b_{1} ^{2} +c_{1} ^{2} }\sqrt{a_{2} ^{2}+b_{2} ^{2} +c_{2} ^{2} } }$

          =      -2×2    +   (-4×4 )       +       (-4×4)

             $\sqrt{(-2^{2} )+(-4^{2} )+(-4^{2})} \sqrt{(2^{2} )+(4^{2} )+(4^{2})}$

           =       $\frac{-4-16-16}{\sqrt{4+16+16}\sqrt{4+16+16} }$

            = $\frac{-36}{\sqrt{36} \sqrt{36} }$

             = $\frac{-36}{36}$

       Cosθ  = -1

       θ = $cos^{-1}$(-1)

      θ = 180°