The angle between u = x i ^ − 3 j ^ − k ^ and v = 2 x i ^ + x j ^ − k ^ is acute, while the angle between v and the positive y -axis is obtuse. Which values can x 2 take?

Question

The angle between u = x i ^ − 3 j ^ ​ − k ^ and v = 2 x i ^ + x j ^ ​ − k ^ is acute, while the angle between v and the positive y -axis is obtuse. Which values can x 2 take?

TestHub · Accepted Answer

Acuteness gives u ⋅ v = 2 x 2 − 3 x + 1 > 0 , i.e. ( 2 x − 1 ) ( x − 1 ) > 0 . The second condition gives x < 0 . Hence every positive value of x 2 is possible, so all listed options work.

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Options:(select one or more)

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