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To solve this problem, we need to express the vector in terms of Cartesian unit vectors. The vector given is in a 2D coordinate system, and its components along the x-axis and y-axis are shown:

- The horizontal component (x-axis) is 3 units.
- The vertical component (y-axis) is 4 units.

### Step 1: Cartesian Form
In terms of Cartesian unit vectors, the vector can be written as:

\[
\vec{A} = 3\hat{i} + 4\hat{j}
\]

Here:
- \(\hat{i}\) represents the unit vector along the x-axis,
- \(\hat{j}\) represents the unit vector along the y-axis.

This matches the option: **3i + 4j**.

### Step 2: Cross-verifying the other options:
- (3, 4): This represents the vector in component form \((x, y)\), which is also correct.
- \(\langle 3, 4 \rangle\): This is another way of writing the vector in component form using angular brackets, also correct.
- \(5 \, | \, 53.13^\circ\): This represents the magnitude and direction form (polar coordinates). Let's verify it.
  
### Step 3: Polar Form
To convert this vector into polar form:
- The magnitude \(r\) of the vector is calculated using the Pythagorean theorem:

\[
r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

- The angle \(\theta\) with the positive x-axis is calculated using:

\[
\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ
\]

Thus, the polar form is \(5 \, | \, 53.13^\circ\), which is also correct.

### Conclusion:
The correct answers are:
- \( \mathbf{3i + 4j} \) (as per the requested form with unit vectors),
- and the other equivalent forms: \((3, 4)\), \(\langle 3, 4 \rangle\), and \(5 \, | \, 53.13^\circ\).

Let me know if you'd like more details or have any questions! Here are five related questions to further your understanding:

1. How do you convert a vector from Cartesian to polar form?
2. What are the unit vectors in a 3D coordinate system?
3. How do you calculate the angle between two vectors?
4. What is the difference between vector magnitude and direction?
5. How do you add two vectors in Cartesian form?

**Tip**: Always verify both the magnitude and direction when converting between polar and Cartesian coordinates.

Exprese el vector en función de los vectores unitarios cartesianos.

Learn how to express a vector using Cartesian unit vectors. This example explains the components (3, 4), vector magnitude (5), and angle (53.13°) in both Cartesian and polar forms. Suitable for high school students in Grades 9-12.

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