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Question
Solve the following :
Show that the lines x = y, z = 0 and x + y = 0, z = 0 intersect each other. Find the vector equation of the plane determined by them.
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Solution
Given lines are x = y, z = 0 and x + y = 0, z = 0.
It is clear that (0, 0, 0) satisfies both the equations.
∴ the lines intersect at O whose position vector is `bar"0"`
Since z = 0 fr both the lines, both the lines ie in XY-plane.
Hence, we have to find equation oXY-ane.
Z-axis is perpendicular to XY-plane.
∴ normal to XY plane is `hat"k"`.
`"O"(bar"0")` lies on the plane.
By using `bar"r".bar"n" = bar"a".bar"n"`, vecttor equation of the required plane is `bar"r".hat"k" = bar"0".bar"k"`
i.e. `bar"r".hat"k"` = 0.
Hence, the given lines intersect each other and the vector equation of the plane determine by them is `bar"r".hat"k"` = 0.
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