To find the magnitude of vector [tex]\(\vec{A}\)[/tex], we will use the formula for the magnitude of a vector in a plane (2-dimensional space).
Given:
[tex]\[
\vec{A} = 3\hat{i} + 2\hat{j}
\][/tex]
The magnitude of a vector [tex]\(\vec{A} = a\hat{i} + b\hat{j}\)[/tex] is calculated using the formula:
[tex]\[
|\vec{A}| = \sqrt{a^2 + b^2}
\][/tex]
For [tex]\(\vec{A}\)[/tex], [tex]\(a = 3\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
|\vec{A}| = \sqrt{(3)^2 + (2)^2}
\][/tex]
Let's calculate each term inside the square root:
[tex]\[
(3)^2 = 9
\][/tex]
[tex]\[
(2)^2 = 4
\][/tex]
Now, sum these:
[tex]\[
9 + 4 = 13
\][/tex]
Finally, take the square root of the sum:
[tex]\[
|\vec{A}| = \sqrt{13} \approx 3.605551275463989
\][/tex]
Therefore, the magnitude of [tex]\(\vec{A} = 3\hat{i} + 2\hat{j}\)[/tex] is approximately [tex]\(3.605551275463989\)[/tex].
