**Given Information:**
- Vector a, vector b, and vector c are non-zero coplanar vectors.
- a · b = b · c = 0
- |a| = 1 unit
- |c| = 6 units

**To Find:**
- |a · c|

**Solution:**

**1. Dot Product of a and b is Zero (a · b = 0):**
The dot product of two vectors is given by the formula:
a · b = |a| |b| cos θ
Since a · b = 0, it implies that cos θ = 0. This means that the angle between vectors a and b is 90 degrees or they are perpendicular to each other.

**2. Dot Product of b and c is Zero (b · c = 0):**
Similarly, since b · c = 0, the angle between vectors b and c is also 90 degrees, or they are perpendicular to each other.

**3. Vectors a, b, and c are Coplanar:**
Since a · b = 0 and b · c = 0, vectors a, b, and c are coplanar. This means that they lie in the same plane.

**4. Magnitude of Vector a is 1 (|a| = 1):**
The magnitude of a vector can be calculated using the formula:
|a| = sqrt(a₁² + a₂² + a₃²) where a₁, a₂, a₃ are the components of vector a.
Given |a| = 1, we can conclude that a has unit magnitude.

**5. Magnitude of Vector c is 6 (|c| = 6):**
Similarly, using the magnitude formula: |c| = sqrt(c₁² + c₂² + c₃²), we find that |c| = 6.

**6. Relationship between Magnitude and Dot Product:**
The dot product of two vectors can also be calculated using the formula:
a · b = |a| |b| cos θ, where θ is the angle between vectors a and b.
Since a · b = 0, and |a| = 1, it implies that |b| cos θ = 0. As cos θ = 0, it means that either |b| = 0 or cos θ = 0.
But we know that |b| ≠ 0, so it must be cos θ = 0.

**7. Magnitude of a · c (|a · c|):**
Now, we need to find the magnitude of a · c, which can be calculated using the formula:
|a · c| = |a| |c| cos θ, where θ is the angle between vectors a and c.
Since a · b = 0, and |a| = 1, it implies that |c| cos θ = 0. As cos θ = 0, it means that either |c| = 0 or cos θ = 0.
But we know that |c| ≠ 0, so it must be cos θ = 0.

Therefore, |a · c| = |a| |c| cos θ = 1 * 6 * 0 = 0.