Examine examples of vectors including displacement, velocity and force.

A particle moves from point P(2, 3, −1) to point R(8, −7, 9) in 5 seconds. Calculate the average velocity vector (in m/s).

A particle moves from point P(3, −2, 5) to point Q(−1, 4, 2) in 4 seconds, then continues to point R(2, 1, −3) in a further 3 seconds. Determine the average velocity vector (m/s) for the entire journey from P to R.

A cargo plane travels from point P to point Q, then immediately continues to point R. The displacement vector from P to Q is **u** = 3**i** + 7**j** + 2**k** kilometres, and the displacement vector from Q to R is **v** = −5**i** + 2**j** + 4**k** kilometres. a) Determine the total displacement vector from P to R. (1 mark) b) Calculate the magnitude of the total displacement from P to R, correct to two decimal places. (1 mark) c) During the journey from P to Q, the plane takes 0.25 hours. Determine the average velocity vector of the plane during this part of the journey, expressing your answer in the form a**i** + b**j** + c**k** km h⁻¹. (1 mark)

A particle moves in three-dimensional space. At time $t = 0$ seconds, the particle is at position $\mathbf{r}_0 = 3\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}$ metres. At time $t = 4$ seconds, the particle is at position $\mathbf{r}_1 = 11\mathbf{i} + 10\mathbf{j} + 13\mathbf{k}$ metres. a) Determine the displacement vector of the particle from $t = 0$ to $t = 4$ seconds. b) Use your result from part (a) to determine the magnitude of the displacement, correct to two decimal places. c) Determine the average velocity vector of the particle over this time interval.

A drone moves from point P(2, -3, 5) to point Q(7, 1, -2), where coordinates are measured in metres. The drone takes 4.0 seconds to complete this movement. a) Determine the displacement vector $\vec{PQ}$ of the drone. (1 mark) b) Use your result from part a) to determine the average velocity vector of the drone during this motion. Express your answer in component form correct to two decimal places. (2 marks)