To solve this problem, we need to address two parts: finding the magnitude of the vector from the origin to the point [tex]\((-7, -5)\)[/tex] and expressing this vector as the sum of unit vectors.
### Finding the Magnitude
The magnitude of a vector [tex]\(\vec{v} = (x, y)\)[/tex] from the origin (0, 0) to the point [tex]\((x, y)\)[/tex] is given by the formula:
[tex]\[
\text{Magnitude} = \sqrt{x^2 + y^2}
\][/tex]
For the point [tex]\((-7, -5)\)[/tex]:
[tex]\[
\text{Magnitude} = \sqrt{(-7)^2 + (-5)^2}
\][/tex]
Calculating inside the square root:
[tex]\[
(-7)^2 = 49
\][/tex]
[tex]\[
(-5)^2 = 25
\][/tex]
Then, add these values:
[tex]\[
49 + 25 = 74
\][/tex]
Thus, the magnitude is:
[tex]\[
\sqrt{74} \approx 8.602325267042627
\][/tex]
### Expressing the Vector as the Sum of Unit Vectors
The vector from the origin to the point [tex]\((-7, -5)\)[/tex] can be written in terms of the unit vectors [tex]\(\vec{i}\)[/tex] (the unit vector in the x-direction) and [tex]\(\vec{j}\)[/tex] (the unit vector in the y-direction) as:
[tex]\[
\vec{v} = -7\vec{i} + (-5\vec{j}) = -7\vec{i} - 5\vec{j}
\][/tex]
### Evaluating the Given Choices
Now, we compare our results with the given choices:
1. a. [tex]\(\sqrt{74}, 7 \vec{i} + 5 \vec{j}\)[/tex]
2. b. [tex]\(\sqrt{75}, 7 \vec{i} + 5 \vec{j}\)[/tex]
3. c. [tex]\(\sqrt{74}, -7 \vec{i} - 5 \vec{j}\)[/tex]
4. d. [tex]\(\sqrt{75}, -7 \vec{i} - 5 \vec{j}\)[/tex]
From our calculations, we find:
- The magnitude of the vector is [tex]\(\sqrt{74}\)[/tex].
- The vector expressed as the sum of unit vectors is [tex]\(-7\vec{i} - 5\vec{j}\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\text{C}}
\][/tex]
