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Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Q 2     Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Answers (1)
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We have given points where the line is passing through it;

Consider the line joining the points (1, – 1, 2) and (3, 4, – 2) is AB and line joining the points  (0, 3, 2) and (3, 5, 6).is CD.

So, we will find the direction ratios of the lines AB and CD;

Direction ratios of AB are a_{1},b_{1}, c_{1}

(3-1),\ (4-(-1)),\ and\ (-2-2)   or  2,\ 5,\ and\ -4

Direction ratios of CD are a_{2},b_{2}, c_{2}

(3-0),\ (5-3)),\ and\ (6-2)   or  3,\ 2,\ and\ 4.

Now, lines AB and CD will be perpendicular to each other if a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2} =0

a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2} =\left ( 2\times3 \right ) +\left ( 5\times2 \right )+ \left ( -4\times 4 \right )

= 6+10-16 = 0

Therefore, AB and CD are perpendicular to each other.

 

 

 

 

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