Vx=−4.5cos60∘+3.0, Vy=4.5sin60∘
solve into magnitude and direction

To solve for the magnitude and direction of a vector given its components \( V_x \) and \( V_y \), we can follow these steps:

### 1. Break down the components:
- \( V_x = -4.5 \cos 60^\circ + 3.0 \)
- \( V_y = 4.5 \sin 60^\circ \)

Now, let's calculate these values.

### Step 1: Calculate \( V_x \)
\( V_x = -4.5 \cos 60^\circ + 3.0 \)

Since \( \cos 60^\circ = 0.5 \):
\[
V_x = -4.5 \times 0.5 + 3.0 = -2.25 + 3.0 = 0.75
\]

### Step 2: Calculate \( V_y \)
\( V_y = 4.5 \sin 60^\circ \)

Since \( \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866 \):
\[
V_y = 4.5 \times 0.866 = 3.897
\]

### 2. Find the magnitude of the vector:
The magnitude of the vector \( V \) is given by:
\[
|V| = \sqrt{V_x^2 + V_y^2}
\]
Substituting the values of \( V_x \) and \( V_y \):
\[
|V| = \sqrt{(0.75)^2 + (3.897)^2} = \sqrt{0.5625 + 15.18} = \sqrt{15.7425} \approx 3.969
\]

### 3. Find the direction of the vector:
The direction \( \theta \) of the vector is given by:
\[
\theta = \tan^{-1}\left(\frac{V_y}{V_x}\right)
\]
Substituting the values of \( V_x \) and \( V_y \):
\[
\theta = \tan^{-1}\left(\frac{3.897}{0.75}\right) = \tan^{-1}(5.196) \approx 79.04^\circ
\]

### Final Answer:
- **Magnitude**: \( |V| \approx 3.97 \)
- **Direction**: \( \theta \approx 79.04^\circ \) (above the positive x-axis)

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Would you like more details or have any questions?

Here are 5 related questions:
1. How does vector addition affect magnitude and direction?
2. Can you find the resultant of two vectors with different angles?
3. How would the vector change if \( V_x \) were negative?
4. What happens to the magnitude if the vector's components are scaled by a factor?
5. How is the direction of a vector determined in different quadrants?

**Tip**: The magnitude of a vector always remains positive, while its direction depends on the signs of \( V_x \) and \( V_y \).

Vx=−4.5cos60∘+3.0, Vy=4.5sin60∘ solve into magnitude and direction

Learn how to solve for the magnitude and direction of a vector with components Vx=−4.5cos60°+3.0 and Vy=4.5sin60°. This step-by-step solution covers trigonometry concepts such as vector components, magnitude, and direction. Suitable for high school students in Grades 10-12.

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