<Question>
A $90^\circ$ rotation around the origin in the counter-clockwise direction is applied to $7 + 2i.$  What is the resulting complex number?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out what happens when I rotate the complex number 7 + 2i by 90 degrees counter-clockwise around the origin. Hmm, let me think. I remember that rotations in the complex plane can be done by multiplying by a complex number of unit magnitude. Since a 90-degree rotation is a common one, maybe there's a standard way to handle that.

First, let me recall that multiplying a complex number by i corresponds to a 90-degree counter-clockwise rotation.
For example, if I have a complex number z = a + bi, then multiplying by i gives i*z = -b + ai, which is a rotation by 90 degrees. Let me verify that with an example.

<prompt_1.1> Verify the 90-degree rotation by multiplying 1 + 0i by i and checking the result. </prompt_1.1>

Another way to represent rotations is using the formula: if you have a point (x, y) and you want to rotate it by an angle θ around the origin, the new point (x', y') is given by:

x' = x*cosθ - y*sinθ
y' = x*sinθ + y*cosθ

Since this is a counter-clockwise rotation, the angle θ is positive. For θ = 90 degrees, cosθ is 0 and sinθ is 1.
Plugging those into the formula:

x' = x*0 - y*1 = -y
y' = x*1 + y*0 = x

So, the new coordinates after rotation would be (-y, x). Translating that back to a complex number, if the original number is x + yi, the rotated number is -y + xi, which is the same as multiplying by i.
Let me confirm that. If original z = x + yi, then i*z = i*x + i*yi = xi + y(i^2) = xi - y = -y + xi.

So, both methods agree. Therefore, applying a 90-degree counter-clockwise rotation to a complex number is equivalent to multiplying the number by i.
Therefore, for the complex number 7 + 2i, multiplying by i should give the result.
Let me do that multiplication step by step.

<prompt_1.2> Multiply the complex number 7 + 2i by i and calculate the result. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Take the complex number 1 + 0i. Multiplying by i gives 0 + 1i, which is a rotation of 90 degrees up (counter-clockwise) from the real axis to the imaginary axis. That makes sense.
</execution_1.1>

<execution_1.2>
Multiplying 7 + 2i by i: 

i*(7 + 2i) = 7*i + 2i*i = 7i + 2i^2.

But i^2 is equal to -1, so 2i^2 = 2*(-1) = -2. Therefore, the result is 7i - 2, which can be written as -2 + 7i.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.2:
Wait, let me check that again.
The multiplication: i*(7 + 2i) = 7i + 2i^2.

<prompt_2.1> Verify the multiplication result by breaking down the steps again. </prompt_2.1>

Alternatively, using the coordinate rotation formula: the original complex number 7 + 2i corresponds to the point (7, 2).
Applying the rotation formulas:

<prompt_2.2> Apply the rotation formula to the point (7, 2) and verify the result. </prompt_2.2>

So, converting back to a complex number, -2 + 7i.
Therefore, the result should be -2 + 7i.

Let me visualize this to make sure. The original point is in the first quadrant, since both real and imaginary parts are positive.

<prompt_2.3> Visualize the rotation of the complex number 7 + 2i by 90 degrees counter-clockwise. </prompt_2.3>

Wait, but sometimes I get confused with directions. Let me confirm once more.

<prompt_2.4> Confirm the rotation direction and result by reviewing the multiplication and rotation steps. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Then, 7i is straightforward, and 2i^2 is 2*(-1) = -2. So combining the terms: 7i - 2. Writing it in standard form for a complex number, that's -2 + 7i.
</execution_2.1>

<execution_2.2>
x = 7, y = 2.

After rotation, x' = -y = -2, y' = x = 7. So the new coordinates are (-2, 7), which corresponds to the complex number -2 + 7i.
</execution_2.2>

<execution_2.3>
A 90-degree counter-clockwise rotation should move it to the second quadrant, where the real part is negative and the imaginary part is positive. The coordinates after rotation would be (-2, 7), which corresponds to the complex number -2 + 7i.
</execution_2.3>

<execution_2.4>
If I rotate 7 + 2i (which is at (7,2)) 90 degrees counter-clockwise, the new point should be (-2,7). Because the formula says x' = -y and y' = x. So original x is 7, original y is 2. Therefore, new x is -2, new y is 7. So the complex number is -2 + 7i.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2, execution_2.3, execution_2.4:
Alternatively, if I think of the vector [7, 2], rotating it 90 degrees counter-clockwise would transform it to [-2, 7].
Which matches the result we have. So, all methods point to -2 + 7i.

Therefore, I think the answer is correct.
I don't see any mistakes in the reasoning. Both algebraic multiplication and geometric coordinate transformation give the same result.
</Plan_3>

<Execution_3>
No prompt to execute.
</Execution_3>

<Final_answer>
Therefore, the resulting complex number after a $90^\circ$ counter-clockwise rotation around the origin is $\boxed{-2 + 7i}$.
</Final_answer>
</think>