Questions: 18. Points A and B, are shown in the figure below within a 3D Cartesian coordinate system. Point B is located on the y-z plane. Answer the following questions. (a) Write a Cartesian position vector representing the position of point A with respect to point B. (b) Calculate the distance between the two points.

Solution Steps

• Point A is located at (2, 4, -6) in the 3D Cartesian coordinate system.
• Point B is located at (0, 5, 3) in the 3D Cartesian coordinate system.

• The position vector from B to A is found by subtracting the coordinates of B from the coordinates of A: \[ \vec{r}_{BA} = (2 - 0) \hat{i} + (4 - 5) \hat{j} + (-6 - 3) \hat{k} \] \[ \vec{r}_{BA} = 2 \hat{i} - 1 \hat{j} - 9 \hat{k} \]

• The distance \(d\) between points A and B is given by the magnitude of the position vector \(\vec{r}_{BA}\): \[ d = \sqrt{(2)^2 + (-1)^2 + (-9)^2} \] \[ d = \sqrt{4 + 1 + 81} \] \[ d = \sqrt{86} \] \[ d \approx 9.27 \, \text{m} \]

Final Answer