To determine the unit vector corresponding to a given vector, we must follow several steps. Let's break down the process using the given vector [tex]\( \vec{A} = 5 \hat{i} + 5 \hat{j} \)[/tex]:
1. Calculate the magnitude of [tex]\( \vec{A} \)[/tex]:
The magnitude of a vector [tex]\( \vec{A} \)[/tex] represented as [tex]\( A_i \hat{i} + A_j \hat{j} \)[/tex] is given by:
[tex]\[
|\vec{A}| = \sqrt{A_i^2 + A_j^2}
\][/tex]
For [tex]\( \vec{A} = 5 \hat{i} + 5 \hat{j} \)[/tex]:
[tex]\[
|\vec{A}| = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 7.0710678118654755
\][/tex]
2. Calculate the components of the unit vector:
A unit vector [tex]\( \hat{A} \)[/tex] in the direction of [tex]\( \vec{A} \)[/tex] is obtained by dividing each component of [tex]\( \vec{A} \)[/tex] by the magnitude of [tex]\( \vec{A} \)[/tex]:
[tex]\[
\hat{A}_i = \frac{A_i}{|\vec{A}|}, \quad \hat{A}_j = \frac{A_j}{|\vec{A}|}
\][/tex]
Thus, for [tex]\( \vec{A} = 5 \hat{i} + 5 \hat{j} \)[/tex]:
[tex]\[
\hat{A}_i = \frac{5}{7.0710678118654755} = 0.7071067811865475
\][/tex]
[tex]\[
\hat{A}_j = \frac{5}{7.0710678118654755} = 0.7071067811865475
\][/tex]
3. Formulate the unit vector:
Then, the unit vector [tex]\( \hat{A} \)[/tex] can be written as:
[tex]\[
\hat{A} = 0.7071067811865475 \hat{i} + 0.7071067811865475 \hat{j}
\][/tex]
4. Match the unit vector to the given options:
Looking at the given options:
- (1) [tex]\( \hat{i} + \hat{j} \)[/tex]
- (2) [tex]\( \frac{\hat{i} + \hat{j}}{\sqrt{2}} \)[/tex]
- (3) [tex]\( \frac{4}{5} \hat{i} + \frac{3}{5} \hat{j} \)[/tex]
- (4) [tex]\( \frac{\hat{i}}{2} + \frac{\hat{j}}{3} \)[/tex]
Since [tex]\( \frac{1}{\sqrt{2}} \approx 0.7071067811865475 \)[/tex], the correct option that matches our unit vector [tex]\( \hat{A} \)[/tex] is:
[tex]\[
\frac{\hat{i} + \hat{j}}{\sqrt{2}}
\][/tex]
Therefore, the correct answer is:
[tex]\( \boxed{\frac{\hat{i} + \hat{j}}{\sqrt{2}}} \)[/tex]
