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 or perpendicular to another vector, we can use vector projections. The projection of one vector onto another represents the component of the first vector that lies in the same direction as the second vector.
a) To find the component of F parallel to v, we need to calculate the vector projection of F onto v. The formula for the vector projection of a vector A onto another vector B is given by:
projB(A) = (A · ẑ) * ẑ
where "·" denotes the dot product and ẑ is the unit vector in the direction of B.
First, let's calculate the dot product of F and v:
F · v = (-21j) · (4i - j)
= 0 - (-21)(1) [since j · i = 0 and j · j = |j|^2 = 1]
= 21
Next, let's calculate the unit vector ẑ in the direction of v:
|v| = sqrt((4)^2 + (-1)^2)
= sqrt(16 + 1)
= sqrt(17)
ẑ = v / |v|
= (4i - j) / sqrt(17)
Now, we can calculate the component of F parallel to v:
projv(F) = (F · ẑ) * ẑ
= (21) * ((4i - j) / sqrt(17))
= (84i - 21j) / sqrt(17)
Therefore, the component of F parallel to v is (84i - 21j) / sqrt(17).
b) To find the component of F perpendicular to v, we can subtract the component of F parallel to v from F:
F_perpendicular = F - projv(F)
= (-21j) - (84i - 21j) / sqrt(17)
= (0i - 21j) - (84i - 21j) / sqrt(17)
= (-84i + 0j) / sqrt(17)
= -84i / sqrt(17)
Therefore, the component of F perpendicular to v is -84i / sqrt(17).