The force on an object is F =−21j . For the vector v =4i −j , find:
(a) The component of F parallel to v :
(b) The component of F perpendicular to v :
To find the component of a vector parallel to another vector, we need to use the concept of the dot product. The dot product of two vectors, A and B, is given by the formula:
A · B = |A| |B| cos θ
where |A| and |B| are the magnitudes of vectors A and B, respectively, and θ is the angle between them. In this case, we need to find the component of F parallel and perpendicular to v, so let's calculate it step by step.
(a) The component of F parallel to v:
To find the component of F parallel to v, we need to project F onto the direction of v. The formula for the projection of a vector A onto another vector B is:
Proj A (B) = (A · B / |B|²) B
First, let's calculate the dot product of F and v:
F · v = (-21j) · (4i - j)
= -21(0) + (-1)(-1)
= 1
Now, let's calculate the magnitude of v:
|v| = sqrt((4)^2 + (-1)^2)
= sqrt(16 + 1)
= sqrt(17)
Using these values, we can calculate the projection of F onto v:
Proj F (v) = (F · v / |v|²) v
= (1 / (sqrt(17))^2) (4i - j)
= (1 / 17) (4i - j)
= (4/17)i - (1/17)j
So, the component of F parallel to v is (4/17)i - (1/17)j.
(b) The component of F perpendicular to v:
To find the component of F perpendicular to v, we subtract the parallel component of F from F. So:
F perpendicular = F - (F parallel)
= -21j - [(4/17)i - (1/17)j]
= -21j - (4/17)i + (1/17)j
= -(4/17)i - (16/17)j
So, the component of F perpendicular to v is -(4/17)i - (16/17)j.