Step-by-step explanation:
To find the displacement of the particle, we need to subtract the initial position vector from the final position vector.
Given:
Initial position vector = (5, 2)
Final position vector = (-2, -3)
a) Displacement vector = Final position vector - Initial position vector
Displacement vector = (-2, -3) - (5, 2)
Displacement vector = (-2 - 5, -3 - 2)
Displacement vector = (-7, -5)
Therefore, the displacement of the particle is (-7, -5).
b) To find the magnitude and direction of the displacement, we can use the Pythagorean theorem and trigonometry.
Magnitude of the displacement vector (|d|) can be calculated using the formula:
|d| = √(dx^2 + dy^2)
Substituting the values:
|d| = √((-7)^2 + (-5)^2)
|d| = √(49 + 25)
|d| = √74
The direction of the displacement vector (θ) can be calculated using the formula:
θ = tan^(-1)(dy / dx)
Substituting the values:
θ = tan^(-1)(-5 / -7)
θ = tan^(-1)(5/7)
Therefore, the magnitude of the displacement is approximately √74 and the direction is approximately tan^(-1)(5/7).