1.

Determine the sine of the angle between the vectors . `2 hat I + 3 hat j -4 hat k` and ( hat i-2 hat k).

Answer» Here `vec A =( 2 hat I + 3 hat j -4 hat k)`
and `vec B =(hat I -2 hat j + 2 hat k)`
` (vec A xx vec B) =|(hat I, hat j, hat k), (2, 3, -4),(1, -2, 2)|`
`=hat I (6-8 ) + hat j (-4 -4 ) + hat k (-4 -3)`
`=- 2 hat I -8 hat j-7 hat k`
`| vec A xx vec B| = sqrt ((-2)^(2) + (-8)^(2) + (-7) ^(2)) = sqrt (29),`
`| vec B | = sqrt (1^(2) + (-2)^(2) + 2^(2)) =sqrt 9`
`sin theta = (|vec A xx vec B|)/(|vec A | | vec B|) =(sqrt117)/(sqrt 29 sqrt9) =(sqrt 23)/(29)`
` or `theta = sin^(-1) sqrt (13)/(29)`.


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