Question

Find the unit vector of direction for the following vector quantities:

(a) Force →F=(3.0ˆi−2.0ˆj)N
a) ˆF = (3.0/√13ˆi - 2.0/√13ˆj)
b) ˆF = (3.0/5ˆi - 2.0/5ˆj)
c) ˆF = (-3.0/√13ˆi + 2.0/√13ˆj)
d) ˆF = (-3.0/5ˆi + 2.0/5ˆj)

(b) Displacement →D=(−3.0ˆi−4.0ˆj)m
a) ˆD = (-3.0/5ˆi - 4.0/5ˆj)
b) ˆD = (-3.0/√25ˆi - 4.0/√25ˆj)
c) ˆD = (3.0/5ˆi + 4.0/5ˆj)
d) ˆD = (3.0/√25ˆi + 4.0/√25ˆj)

(c) Velocity →v=(−5.00ˆi+4.00ˆj)m/s
a) ˆv = (-5.00/√41ˆi + 4.00/√41ˆj)
b) ˆv = (-5.00/√81ˆi + 4.00/√81ˆj)
c) ˆv = (5.00/√41ˆi - 4.00/√41ˆj)
d) ˆv = (5.00/√81ˆi - 4.00/√81ˆj)

Answer 1

The unit vector for a given vector quantity is found by dividing each component by the magnitude of the vector. For the given vectors (force, displacement, velocity), the unit vectors are (3.0/√13ˆi - 2.0/√13ˆj), (-3.0/5ˆi - 4.0/5ˆj), and (-5.00/√41ˆi + 4.00/√41ˆj) respectively.

Step-by-step explanation:

To find the unit vector of a given vector, we need to divide each component of the vector by its magnitude. The magnitude of a vector →V = (Vx, Vy, Vz) can be calculated using the formula |→V| = √(Vx² + Vy² + Vz²).

For example, for the force vector →F=(3.0ˆi - 2.0ˆj)N, we first find its magnitude using the formula for magnitude of a two-dimensional vector, which is:

|→F| = √(3.0² + (-2.0)²) = √(9 + 4) = √13

Then, the unit vector of the force is:

ˆF = (3.0/√13ˆi - 2.0/√13ˆj)

Following the same procedure for the displacement vector →D=(−3.0ˆi - 4.0ˆj)m and the velocity vector →v=(−5.00ˆi + 4.00ˆj)m/s, we obtain their unit vectors respectively:

ˆD = (-3.0/√25ˆi - 4.0/√25ˆj)

ˆv = (-5.00/√41ˆi + 4.00/√41ˆj)