If a unit vector makes an angles with with and an acute angle θ with , then find θ and hence, the compounds of .
It is given that unit vector makes angle π 3 with i ^ and angle π 4 with j ^ and an acute angle θ with k ^ .
Let, components of unit vector ( a → ) is ( a 1 , a 2 , a 3 ) .
a → = a 1 i ^ + a 2 j ^ + a 3 k ^
Unit vector makes an angle of π 3 with i ^ ,
cos π 3 = a 1 | a → | 1 2 = a 1
Unit vector makes an angle of π 4 with j → ,
cos π 4 = a 2 | a → | 1 2 = a 2
Unit vector makes an acute angle θ with k ^ ,
cos θ = a 3 | a → | cos θ = a 3
Since magnitude of a → is 1,
a 1 2 + a 2 2 + a 3 2 = 1 ( 1 2 ) 2 + ( 1 2 ) 2 + cos 2 θ = 1 3 4 + cos 2 θ + 1 2 = 1 cos 2 θ = 1 − 3 4
Further solve.
cos 2 θ = 1 4 cos θ = 1 2 θ = π 3
Again solve,
a 3 = cos 2 θ = cos 2 π 3 = 1 2
Thus, components of a → are ( 1 2 , 1 2 , 1 2 ) and θ is π 3 .
It is given that unit vector makes angle
Let, components of unit vector
Unit vector makes an angle of
Unit vector makes an angle of
Unit vector makes an acute angle
Since magnitude of
Further solve.
Again solve,
Thus, components of
makes an angles
with
with
and
an acute angle θ with
,
then find θ and hence, the compounds of
.