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To find the angle between the vector \( \vec{v} = 15\hat{i} + 15\hat{j} \) and the positive \( x \)-axis, we can use the formula:

\[
\theta = \tan^{-1} \left( \frac{y}{x} \right)
\]

Here, \( x = 15 \) and \( y = 15 \), as the vector is written in the form \( \vec{v} = x\hat{i} + y\hat{j} \).

### Step-by-step solution:

1. Substitute the values of \( x \) and \( y \):
   \[
   \theta = \tan^{-1} \left( \frac{15}{15} \right) = \tan^{-1}(1)
   \]

2. The inverse tangent of 1 is a known value:
   \[
   \theta = 45^\circ
   \]

Thus, the angle between the vector and the positive \( x \)-axis is \( 45^\circ \) measured anticlockwise.

If you need further details or have any questions, feel free to ask!

### Relative questions:
1. How do you compute the magnitude of a vector like \( 15\hat{i} + 15\hat{j} \)?
2. What is the physical meaning of the angle between a vector and the \( x \)-axis?
3. Can you calculate the angle between any two given vectors?
4. How would the result change if one of the components of the vector were negative?
5. What is the relationship between the angle and the slope of a vector?

### Tip:
Always ensure you're using the correct quadrant when applying the inverse tangent, especially when dealing with negative components.

Find the angle between the vector 15i + 15j and the positive x-axis. Give your answer correct to 1 decimal place.

Learn how to find the angle between the vector 15i + 15j and the positive x-axis. This problem uses trigonometry concepts like the inverse tangent function, and it's suitable for students in Grades 9-12. The solution is explained with a step-by-step approach.

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