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To find the magnitude of vector \( \mathbf{V} \) when its horizontal component \( V_x \) and vertical component \( V_y \) are given, we use the Pythagorean theorem. The formula for the magnitude of the vector is:

\[
|\mathbf{V}| = \sqrt{V_x^2 + V_y^2}
\]

Given:
- \( |V_x| = 40.0 \)
- \( |V_y| = 23.0 \)

Substituting these values into the formula:

\[
|\mathbf{V}| = \sqrt{(40.0)^2 + (23.0)^2}
\]

Let me calculate that.The magnitude of vector \( \mathbf{V} \) is approximately \( 46.1 \), rounded to one decimal place.

Would you like more details or have any questions?

Here are five related questions for further practice:
1. How do you determine the direction of vector \( \mathbf{V} \) based on its components?
2. What would the magnitude of \( \mathbf{V} \) be if \( |V_x| = 50 \) and \( |V_y| = 30 \)?
3. Can you express vector \( \mathbf{V} \) in terms of unit vectors \( \hat{i} \) and \( \hat{j} \)?
4. What is the relationship between the angle of the vector and its components?
5. How do the components of a vector change if it is rotated by 90 degrees?

**Tip:** The magnitude of a vector is always non-negative, but the direction of the vector can vary based on the sign of its components.

For the problem below, the magnitudes of the horizontal and vertical vector components, Vx and Vy of vector V are given. In this case, find the magnitude of V. Given |Vx| = 40.0 and |Vy| = 23.0. Round your answer to one decimal place.

Learn how to find the magnitude of a vector given its horizontal and vertical components. In this example, |Vx| = 40.0 and |Vy| = 23.0, and the solution involves applying the Pythagorean Theorem to calculate |V|.

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