Let's solve each part step-by-step for the given vectors:
We have:
[tex]\[
\begin{array}{l}
\bar{r}_1=3 i -2 j + k \\
\bar{r}_2=2 i -4 j -3 k \\
\bar{r}_3=-i +2 j +2 k
\end{array}
\][/tex]
### Part (a): Magnitude of [tex]\(\bar{r}_3\)[/tex]
The magnitude of a vector [tex]\(\vec{v} = a i + b j + c k\)[/tex] is given by:
[tex]\[
\|\vec{v}\| = \sqrt{a^2 + b^2 + c^2}
\][/tex]
For [tex]\(\bar{r}_3\)[/tex]:
[tex]\[
\bar{r}_3 = -i + 2j + 2k
\][/tex]
[tex]\[
\|\bar{r}_3\| = \sqrt{(-1)^2 + 2^2 + 2^2}
\][/tex]
Calculating the components:
[tex]\[
\|\bar{r}_3\| = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\][/tex]
So, the magnitude of [tex]\(\bar{r}_3\)[/tex] is:
[tex]\[
\boxed{3}
\][/tex]
### Part (b): Magnitude of [tex]\(\bar{r}_1 + \bar{r}_2 + \bar{r}_3\)[/tex]
First, we need to find the vector sum of [tex]\(\bar{r}_1 + \bar{r}_2 + \bar{r}_3\)[/tex]:
[tex]\[
\bar{r}_1 = 3i - 2j + k
\][/tex]
[tex]\[
\bar{r}_2 = 2i - 4j - 3k
\][/tex]
[tex]\[
\bar{r}_3 = -i + 2j + 2k
\][/tex]
Adding the vectors:
[tex]\[
\bar{r}_1 + \bar{r}_2 + \bar{r}_3 = (3i + 2i - i) + (-2j - 4j + 2j) + (k - 3k + 2k)
\][/tex]
[tex]\[
= (3 + 2 - 1)i + (-2 - 4 + 2)j + (1 - 3 + 2)k
\][/tex]
[tex]\[
= 4i - 4j + 0k
\][/tex]
[tex]\[
= 4i - 4j
\][/tex]
So, the sum vector is:
[tex]\[
\bar{r}_1 + \bar{r}_2 + \bar{r}_3 = 4i - 4j
\][/tex]
Now, we find the magnitude of the sum vector:
[tex]\[
\|\bar{r}_1 + \bar{r}_2 + \bar{r}_3\| = \sqrt{4^2 + (-4)^2}
\][/tex]
[tex]\[
= \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.656854249492381
\][/tex]
So, the magnitude of [tex]\(\bar{r}_1 + \bar{r}_2 + \bar{r}_3\)[/tex] is:
[tex]\[
\boxed{5.656854249492381}
\][/tex]
### Part (c): Magnitude of [tex]\(2 \bar{r}_1 - 3 \bar{r}_2 - 5 \bar{r}_3\)[/tex]
First, let's find the new vector:
[tex]\[
2 \bar{r}_1 = 2(3i - 2j + k) = 6i - 4j + 2k
\][/tex]
[tex]\[
-3 \bar{r}_2 = -3(2i - 4j - 3k) = -6i + 12j + 9k
\][/tex]
[tex]\[
-5 \bar{r}_3 = -5(-i + 2j + 2k) = 5i - 10j - 10k
\][/tex]
Adding these:
[tex]\[
2 \bar{r}_1 - 3 \bar{r}_2 - 5 \bar{r}_3 = (6i - 4j + 2k) + (-6i + 12j + 9k) + (5i - 10j - 10k)
\][/tex]
[tex]\[
= (6 - 6 + 5)i + (-4 + 12 - 10)j + (2 + 9 - 10)k
\][/tex]
[tex]\[
= 5i - 2j + 1k
\][/tex]
So, the new vector is:
[tex]\[
2 \bar{r}_1 - 3 \bar{r}_2 - 5 \bar{r}_3 = 5i - 2j + k
\][/tex]
Now, we find the magnitude of this new vector:
[tex]\[
\|2 \bar{r}_1 - 3 \bar{r}_2 - 5 \bar{r}_3\| = \sqrt{5^2 + (-2)^2 + 1^2}
\][/tex]
[tex]\[
= \sqrt{25 + 4 + 1} = \sqrt{30} \approx 5.477225575051661
\][/tex]
So, the magnitude of [tex]\(2 \bar{r}_1 - 3 \bar{r}_2 - 5 \bar{r}_3\)[/tex] is:
[tex]\[
\boxed{5.477225575051661}
\][/tex]
