https://file.aiquickdraw.com/mathbot/file/1727990375435lmnml8yh.jpg

The image represents a vector equation:

\[
\vec{v} = 7\hat{i} + 3\hat{j}
\]

This vector is expressed in two dimensions (2D), where:

- \( \hat{i} \) is the unit vector along the x-axis, and
- \( \hat{j} \) is the unit vector along the y-axis.

The components of the vector \( \vec{v} \) are:

- \( 7 \) in the x-direction, and
- \( 3 \) in the y-direction.

Thus, the vector \( \vec{v} \) can be written as \( \vec{v} = (7, 3) \) in component form.

### Magnitude of the Vector:
The magnitude (or length) of the vector can be found using the Pythagorean theorem:

\[
|\vec{v}| = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58} \approx 7.62
\]

### Direction of the Vector:
The direction (angle \( \theta \)) of the vector relative to the positive x-axis can be found using the tangent function:

\[
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{3}{7}\right) \approx 23.2^\circ
\]

Let me know if you want further details or have any questions.

### Follow-up Questions:
1. How do you calculate the dot product between two vectors?
2. What is the difference between a scalar and vector quantity?
3. How do you find the unit vector in the direction of \( \vec{v} \)?
4. What is the physical significance of a vector’s magnitude?
5. Can you calculate the cross product in two dimensions?

#### Tip:
The magnitude of a vector gives an idea of its size, while its direction shows where it points relative to a reference axis.

Find the magnitude and direction of the vector \( \vec{v} = 7\hat{i} + 3\hat{j} \)

Learn how to find the magnitude and direction of the vector \( 7\hat{i} + 3\hat{j} \). This example explains vector algebra, the Pythagorean theorem, and how to calculate the direction using trigonometry. Suitable for high school students.

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