The force on an object is F=−20j. For the vector v=−5i−4j, 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 force F parallel to vector v, we need to use the dot product. The dot product of two vectors A and B is given by
A · B = |A| |B| cos(θ),
where |A| represents the magnitude of vector A, |B| represents the magnitude of vector B, and θ represents the angle between A and B.
(a) The component of F parallel to v:
To find the component of F parallel to v, we can use the formula:
F_parallel = F · (v̂)
where v̂ is the unit vector in the direction of v.
First, we need to find the magnitude of vector v:
|v| = √((-5)^2 + (-4)^2) = √(25 + 16) = √41.
Next, we need to find the unit vector in the direction of v:
v̂ = v / |v| = (-5i - 4j) / √41.
Now, we can compute the dot product of F and v̂:
F_parallel = F · v̂ = (-20j) · (-5i - 4j) / √41.
To calculate the dot product, we multiply the corresponding components of the vectors and add the results:
F_parallel = (-20) · (-4) / √41 = 80 / √41.
Therefore, the component of F parallel to v is 80 / √41.
(b) The component of F perpendicular to v:
To find the component of F perpendicular to v, we can use the formula:
F_perpendicular = F - F_parallel.
First, let's substitute the value of F_parallel we found in part (a):
F_perpendicular = -20j - (80 / √41).
Thus, the component of F perpendicular to v is -20j - (80 / √41).
(c) The work, W, done by force F through displacement v:
The work done by force F through displacement v is given by the formula:
W = F · v.
We can now calculate the dot product of F and v:
W = F · v = (-20j) · (-5i - 4j).
To calculate the dot product, we multiply the corresponding components of the vectors and add the results:
W = (-20) · (-4) = 80.
Therefore, the work done by force F through displacement v is 80.