Vector Rotation - Casting a Cone

Question

I'm working on an obstacle avoidance system in unity, and my thought is to cast a sphere along the velocity; if it hits, draw a ring of spheres around that one and repeat until max ring count or a clear path is found.

2D example

To accomplish this, I'm currently rotating the vector up and then rotating again using the velocity as the axis of rotation, and casting a sphere along the new vector.

working example

This works fine (although I'm not sure how performant it is) until the initial direction is not one of the cardinal directions.

what's happening when things go wrong

My understanding is that after the following line, rotatedVector would be a vector of equal magnitude to ray, but pointing in a new direction that could be described as having rotated about the velocityDirection a number of degrees equal to the magnitude of velocityDirection.

var rotatedVector = Quaternion.Euler(velocityDirection) * ray;

But that only seems to be happening when the velocityDirection is aligned with +- World.up, forward, or right.

Any ideas as to what's happening or how I can accomplish this goal?

Code to DrawGizmos

public int NumRings = 3;
private void OnDrawGizmosSelected()
{
    Vector3 velocityDirection = transform.forward * MaxSpeed;
    NumRings = Mathf.Clamp(NumRings, 0, 6);

    Gizmos.color = Color.cyan;
    for (int i = 0; i < NumRings; i++) {
        Vector3 rotationAxis = Vector3.Cross(velocityDirection, velocityDirection + Vector3.up).normalized * Mathf.Atan(i*2/ MaxSpeed) * Mathf.Rad2Deg;
        Vector3 probe;
        if (rotationAxis.magnitude > 0) {
            probe = Quaternion.Euler(rotationAxis) * velocityDirection;
        } else {
            probe = velocityDirection;
        }

        int numProbes = i == 0 ? 1 : 6*i;
        rotationAxis = velocityDirection.normalized;
        for (int j = 0; j < numProbes; j++) {
            var test = Quaternion.Euler(rotationAxis * 360 / numProbes * j) * probe;
            Gizmos.DrawLine(transform.position, transform.position + test);
            Gizmos.DrawWireSphere(transform.position + test, 1f);
        }
    }
}

Answer 1

@DMGregory pointed out that the vector rotation function I was looking for was Quaternion.AngleAxis.

Changing var test to the below resulted in the desired behavior.

var test = Quaternion.AngleAxis(360 / numProbes * j, rotationAxis) * probe;

Thanks again @DMGregory