Answer:
Step-by-step explanation:
- A plane is oriented in a Cartesian coordinate system such that it makes an angle of ( π / 3 ) with the positive x - axis.
- A force ( F ) is directed along the y-axis as a vector < 0 , - 4 >
- We are to determine the the components of force ( F ) parallel and normal to the defined plane.
- We will denote two unit vectors: ( ) parallel to plane and (
) orthogonal to the defined plane. We will define the two unit vectors in ( x - y ) plane as follows:
- The unit vector ( ) parallel to the defined plane makes an angle of ( 30° ) with the positive y-axis and an angle of ( π / 3 = 60° ) with the x-axis. We will find the projection of the vector onto the x and y axes as follows:
= < cos ( 60° ) , cos ( 30° ) >
- Similarly, the unit vector ( ) orthogonal to plane makes an angle of ( π / 3 ) with the positive x - axis and angle of ( π / 6 ) with the y-axis in negative direction. We will find the projection of the vector onto the x and y axes as follows:
- To find the projection of force ( F ) along and normal to the plane we will apply the dot product formulation:
- The Force vector parallel to the plane ( ) would be:
- Similarly, to find the projection of force ( ) normal to the plane we again employ the dot product formulation with normal unit vector (
) as follows:
- To prove that the projected forces ( ) and (
) are correct we will apply the vector summation of the two orthogonal vector which must equal to the original vector < 0 , - 4 >
.. proven
