Question

problem 5
use the following points in \\(\mathbb{r}^3\\) to answer the questions in parts 1 and 2.
\\(p = (5, -2, 8)\\)
\\(q = (7, 4, 5)\\)

problem 5 - part 1
(fill in the blank) : calculate the displacement vector \\(\overrightarrow{pq}\\).
please write your answer as a linear combination of the standard basis vectors \\(\\{\hat{i}, \hat{j}, \hat{k}\\}\\) in \\(\mathbb{r}^3\\).
\\(\overrightarrow{pq} = \text{}\\)

problem 5 - part 2
(fill in the blank) : calculate the distance between the points
for full credit, please show your calculation.
\\(d(p, q) = ||\overrightarrow{pq}|| = \text{<your answer here>}\\)

Explanation:

Step1: Compute $\overrightarrow{PQ}$ components

$\overrightarrow{PQ} = (7-5)\hat{i} + (4-(-2))\hat{j} + (5-8)\hat{k}$

Step2: Simplify the vector components

$\overrightarrow{PQ} = 2\hat{i} + 6\hat{j} - 3\hat{k}$

Step3: Compute norm of $\overrightarrow{PQ}$

$d(P,Q) = \sqrt{(2)^2 + (6)^2 + (-3)^2}$

Step4: Calculate the final distance

$d(P,Q) = \sqrt{4 + 36 + 9} = \sqrt{49} = 7$

Answer:

$\overrightarrow{PQ} = 2\hat{i} + 6\hat{j} - 3\hat{k}$
$d(P,Q) = 7$