If `A ,B ,C ,D` are four distinct point in space such that `A B` is not perpendicular to `C D` and satisfies ` vec A Bdot vec C D=k(| vec A D|^2+| vec B C|^2-| vec A C|^2=| vec B D|^2),` then find the value of `kdot`

Explanation

To solve the problem, we need to analyze the given vector equation and find the value of \( k \). Let's break it down step by step.

### Step 1: Define the vectors
Let:
- \( \vec{A}B = \vec{b} \)
- \( \vec{C}D = \vec{d} \)
- \( \vec{A}D = \vec{d} - \vec{a} \)
- \( \vec{B}C = \vec{c} - \vec{b} \)
...