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 :
The work, W, done by force F through displacement v :
To find the component of a vector parallel to another vector, we can use the dot product. The formula for the component parallel to a vector is given by:
Component parallel to a vector = (Vector dot Product / Magnitude of the vector) * Unit vector of the vector
Now let's find the answers step-by-step:
a) The component of F parallel to v:
To find the component of F parallel to v, we can use the dot product formula:
Component parallel to v = (F dot v / |v|) * (v / |v|)
Given F = -21j and v = 4i - j:
F dot v = (-21j) dot (4i - j) = -21*0 + 4*(-1) = -4
|v| = sqrt((4)^2 + (-1)^2) = sqrt(16 + 1) = sqrt(17)
v / |v| = (4i - j) / sqrt(17)
Now, let's calculate the component parallel to v:
Component parallel to v = (-4 / sqrt(17)) * (4i - j) / sqrt(17)
= (-16i + 4j) / 17
Therefore, the component of F parallel to v is (-16i + 4j) / 17.
b) The component of F perpendicular to v:
To find the component of F perpendicular to v, we can use the formula:
Component perpendicular to v = F - Component parallel to v
Given F = -21j and the component parallel to v = (-16i + 4j) / 17, we can subtract the two vectors:
Component perpendicular to v = -21j - (-16i + 4j) / 17
= -21j + (16i - 4j) / 17
= (-4j + 16i - 4j) / 17
= (16i - 8j) / 17
Therefore, the component of F perpendicular to v is (16i - 8j) / 17.
c) The work done by force F through displacement v:
The work done by a force through displacement is given by the dot product of the force and the displacement:
W = F dot v
Given F = -21j and v = 4i - j:
W = (-21j) dot (4i - j)
= -21*4 + (-1)(-21)
= -84 + 21
= -63
Therefore, the work done by force F through displacement v is -63.