If a unit vector makes an angle with withand an acute angle θ with , then find the value of θ.
Given the unit vector makes,
• an angle of with x-axis
• an angle of with y-axis
• an angle of θ with z-axis
• θ is acute angle
Let the unit vector be:
As given it is a unit vector,
Therefore = 1
As the angle between in and x-axis is , the scalar product of the vectors can be performed.
The scalar product of the two vectors is given by
[as both the vectors are of magnitude 1].
As the angle between in and y-axis is , the scalar product of the vectors can be performed.
Similarly the angle between in and y-axis is θ , the scalar product of the vectors can be performed.
The magnitude of a vector x+ y+ z is given by .
Now consider the magnitude of the vector
1
1
[Squaring on both sides]
1 =
cos2θ
cos2θ =
cos θ =
cosθ =
As given in the question θ is acute angle, so θ belongs to 1st quadrant and is positive.
Therefore