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Sagot :
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]
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]
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