To determine the length (magnitude) of the vector [tex]\((6, -3)\)[/tex], follow these steps:
1. Identify the components of the vector: The vector is given as [tex]\((6, -3)\)[/tex], where [tex]\(6\)[/tex] is the [tex]\(x\)[/tex]-component and [tex]\(-3\)[/tex] is the [tex]\(y\)[/tex]-component.
2. Use the formula for the magnitude of a vector: The magnitude [tex]\( \| \mathbf{v} \| \)[/tex] of a vector [tex]\(\mathbf{v} = (x, y)\)[/tex] is given by the formula:
[tex]\[
\| \mathbf{v} \| = \sqrt{x^2 + y^2}
\][/tex]
3. Substitute the components into the formula:
[tex]\[
\| (6, -3) \| = \sqrt{6^2 + (-3)^2}
\][/tex]
4. Calculate the squares of the components:
[tex]\[
6^2 = 36
\][/tex]
[tex]\[
(-3)^2 = 9
\][/tex]
5. Add the squared values:
[tex]\[
36 + 9 = 45
\][/tex]
6. Take the square root of the sum:
[tex]\[
\sqrt{45} = 3\sqrt{5}
\][/tex]
Therefore, the magnitude of the vector [tex]\((6, -3)\)[/tex] is [tex]\(3\sqrt{5}\)[/tex].
So, the correct answer is:
C. [tex]\(3\sqrt{5}\)[/tex]
