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Question
The line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.
Options
True
False
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Solution
This statement is False.
Explanation:
Direction ratios of the line `(hat"i" - hat"j" + 2hat"k")`
Direction ratios of the normal to the plane are `(3hat"i" + hat"j" - hat"k")`
So `(hat"i" - hat"j" + 2hat"k").(3hat"i" + hat"j" - hat"k")` = 3 – 1 – 2 = 0
Therefore, the line is parallel to the plane.
Now point through which the line is passing
`vec"a" = 2hat"i" - 3hat"j" - hat"k"`
If line lies in the plane then
`(2hat"i" - 3hat"j" - hat"k").(3hat"i" + hat"j" - hat"k") + 2` = 0
6 – 3 + 1 + 2 ≠ 0
So, the line does not lie in the plane.
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