Question

Consider the points ​P(5​,5​,1​) and ​Q(13​,13​,3​).
a. Find PQ with right arrow and state your answer in two​ forms: (a,b,c) and ai+bj+ck.
b. Find the magnitude of PQ with right arrow.
c. Find two unit vectors parallel to PQ with right arrow.

Answer 1

Answer:

a) $\overrightarrow{PQ} = (8,8, 2)$ or $\overrightarrow{PQ} = 8\,i + 8\,j + 2\,k$, b) The magnitude of segment PQ is approximately 11.489, c) The two unit vectors associated to PQ are, respectively: $\vec v_{1} = (0.696,0.696, 0.174)$ and $\vec v_{2} = (-0.696,-0.696, -0.174)$

Step-by-step explanation:

a) The vectorial form of segment PQ is determined as follows:

$\overrightarrow {PQ} = \vec Q - \vec P$

Where $\vec Q$ and $\vec P$ are the respective locations of points Q and P with respect to origin. If $\vec Q = (13,13,3)$ and $\vec P = (5,5,1)$, then:

$\overrightarrow{PQ} = (13,13,3)-(5,5,1)$

$\overrightarrow {PQ} = (13-5, 13-5, 3 - 1)$

$\overrightarrow{PQ} = (8,8, 2)$

Another form of the previous solution is $\overrightarrow{PQ} = 8\,i + 8\,j + 2\,k$.

b) The magnitude of the segment PQ is determined with the help of Pythagorean Theorem in terms of rectangular components:

$\|\overrightarrow{PQ}\| =\sqrt{PQ_{x}^{2}+PQ_{y}^{2}+PQ_{z}^{2}}$

$\|\overrightarrow{PQ}\| = \sqrt{8^{2}+8^{2}+2^{2}}$

$\|\overrightarrow{PQ}\|\approx 11.489$

The magnitude of segment PQ is approximately 11.489.

c) There are two unit vectors associated to PQ, one parallel and another antiparallel. That is:

$\vec v_{1} = \vec u_{PQ}$ (parallel) and $\vec v_{2} = -\vec u_{PQ}$ (antiparallel)

The unit vector is defined by the following equation:

$\vec u_{PQ} = \frac{\overrightarrow{PQ}}{\|\overrightarrow{PQ}\|}$

Given that $\overrightarrow{PQ} = (8,8, 2)$ and $\|\overrightarrow{PQ}\|\approx 11.489$, the unit vector is:

$\vec u_{PQ} = \frac{(8,8,2)}{11.489}$

$\vec u_{PQ} = \left(\frac{8}{11.489},\frac{8}{11,489},\frac{2}{11.489} \right)$

$\vec u_{PQ} = \left(0.696, 0.696,0.174\right)$

The two unit vectors associated to PQ are, respectively:

$\vec v_{1} = (0.696,0.696, 0.174)$ and $\vec v_{2} = (-0.696,-0.696, -0.174)$