Relative to an origin O, the point A has position vector 4i + 7j - pk and the point B has position vector 8i - j - pk, where p is a constant.
(ii) Find the values of p for which angle AOB = 60◦
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We can find the values of p using the dot product of the two position vectors. By definition,
`veca * vecb = |veca|*|vecb|*cosalpha` , where `alpha` is the angle between the vectors `veca`
and `vecb` .
The dot product of two given vectors and their magnitudes can be found using their coordinates:
Magnitude of the position vector of point A, (4, 7, -p) is
`sqrt(4^2 + 7^2 + (-p)^2) = sqrt(65+p^2)`
Magnitude of the position vector of point B, (8, -1, -p) is
`sqrt(8^2 + (-1)^2 + (-p)^2) = sqrt(65+p^2)`
The dot product of these two vectors is `4*8 + 7*(-1) + (-p)(-p) = 25 + p^2`
Then, cosine between the two vectors can be found as
Plugging in the expressions in terms of p,
`cosalpha=(25+p^2)/(sqrt(65+p^2)sqrt(65+p^2)) = (25+p^2)/(65+p^2)`
If the angle AOB is 60 degrees than its cosine is 1/2
so `(25+p^2)/(65+p^2) = 1/2`
`50+2p^2 = 65+p^2`
`p = +-sqrt15`
Thefore the values of p for which angle AOB is 60 degrees are
`sqrt15` and `-sqrt15` .
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