x = −3.00 cm, y = + 8.00 cm

 

What angle does the vector make with the +y-axis in counterclockwise direction? A positive angle is counterclockwise from the +y-axis.

Given the vector components \(x = -3.00 \, \text{cm}\) and \(y = 8.00 \, \text{cm}\), we need to determine the angle it makes with the positive \(y\)-axis, measured counterclockwise.

### Step-by-Step Process:

1. **Determine the angle with the positive \(x\)-axis**:
   - The tangent of the angle relative to the positive \(x\)-axis is given by:
   \[
   \tan(\theta_x) = \frac{y}{x} = \frac{8.00 \, \text{cm}}{-3.00 \, \text{cm}} = -2.67
   \]
   - Therefore, the angle relative to the \(x\)-axis is:
   \[
   \theta_x = \tan^{-1}(-2.67)
   \]
   Using a calculator:
   \[
   \theta_x \approx -69.44^\circ
   \]

2. **Adjust the angle relative to the positive \(y\)-axis**:
   - The angle we found is measured clockwise from the negative \(x\)-axis, which is equivalent to the angle from the positive \(y\)-axis. The total angle in a full circle is \(90^\circ - \theta_x\), as we are measuring counterclockwise from the \(y\)-axis.
   
   Thus, the angle relative to the \(y\)-axis is:
   \[
   \theta_y = 90^\circ - (-69.44^\circ) = 90^\circ + 69.44^\circ = 159.44^\circ
   \]

Thus, the vector makes an angle of **159.44°** with the positive \(y\)-axis in the counterclockwise direction.

Would you like more details, or do you have any questions?

### Related questions:
1. What would be the angle if the vector had a positive \(x\)-component?
2. How does the quadrant of the vector affect the calculation of the angle?
3. What is the magnitude of the vector with components \((-3.00 \, \text{cm}, 8.00 \, \text{cm})\)?
4. What is the direction of the vector relative to the negative \(y\)-axis?
5. How would the angle change if both components were negative?

**Tip**: Always visualize vector direction when solving trigonometric problems to better understand angle orientation.

x = −3.00 cm, y = + 8.00 cm. What angle does the vector make with the +y-axis in counterclockwise direction? A positive angle is counterclockwise from the +y-axis.

Learn how to calculate the angle made by a vector (x = -3.00 cm, y = 8.00 cm) with the positive y-axis in a counterclockwise direction. This problem covers vector components and trigonometric functions, making it suitable for high school students in Grades 10-12.

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