If `veca, vecb, vecc` are three given non-coplanar vectors and any arbitrary vector `vecr` in space, where `Delta_(1)=|{:(vecr.veca,vecb.veca,vecc.veca),(vecr.vecb,vecb.vecb,vecc.vecb),(vecr.vecc,vecb.vecc,vecc.vecc):}|,Delta_(2)=|{:(veca.veca,vecr.veca,vecc.veca),(veca.vecb,vecr.vecb,vecc.vecb),(veca.vecc,vecr.vecc ,vecc.vecc):}|`
`Delta_(3)=|{:(veca.veca,vecb.veca,vecr.veca),(veca.vecb,vecb.vecb,vecr.vecb),(veca.vecc,vecb.vecc,vecr.vecc):}|, Delta=|{:(veca.veca,vecb.veca,vecc.veca),(veca.vecb,vecb.vecb,vecc.vecb),(veca.vecc,vecb.vecc,vecc.vecc):}|,`` "then prove that " vecr=(Delta_(1))/Deltaveca+(Delta_(2))/Deltavecb+(Delta_(3))/Deltavecc`

Explanation

To prove that \( \vec{r} = \frac{\Delta_1}{\Delta} \vec{a} + \frac{\Delta_2}{\Delta} \vec{b} + \frac{\Delta_3}{\Delta} \vec{c} \), we will follow these steps:

### Step 1: Define the Vectors and Determinants
We have three non-coplanar vectors \( \vec{a}, \vec{b}, \vec{c} \) and an arbitrary vector \( \vec{r} \) in space. The determinants are defined as follows:
- \( \Delta_1 = | \begin{vmatrix} \vec{r} \cdot \vec{a} & \vec{b} \cdot \vec{a} & \vec{c} \cdot \vec{a} \\ \vec{r} \cdot \vec{b} & \vec{b} \cdot \vec{b} & \vec{c} \cdot \vec{b} \\ \vec{r} \cdot \vec{c} & \vec{b} \cdot \vec{c} & \vec{c} \cdot \vec{c} \end{vmatrix} | \)
- \( \Delta_2 = | \begin{vmatrix} \vec{a} \cdot \vec{a} & \vec{r} \cdot \vec{a} & \vec{c} \cdot \vec{a} \\ \vec{a} \cdot \vec{b} & \vec{r} \cdot \vec{b} & \vec{c} \cdot \vec{b} \\ \vec{a} \cdot \vec{c} & \vec{r} \cdot \vec{c} & \vec{c} \cdot \vec{c} \end{vmatrix} | \)
- \( \Delta_3 = | \begin{vmatrix} \vec{a} \cdot \vec{a} & \vec{b} \cdot \vec{a} & \vec{r} \cdot \vec{a} \\ \vec{a} \cdot \vec{b} & \vec{b} \cdot \vec{b} & \vec{r} \cdot \vec{b} \\ \vec{a} \cdot \vec{c} & \vec{b} \cdot \vec{c} & \vec{r} \cdot \vec{c} \end{vmatrix} | \)
- \( \Delta = | \begin{vmatrix} \vec{a} \cdot \vec{a} & \vec{b} \cdot \vec{a} & \vec{c} \cdot \vec{a} \\ \vec{a} \cdot \vec{b} & \vec{b} \cdot \vec{b} & \vec{c} \cdot \vec{b} \\ \vec{a} \cdot \vec{c} & \vec{b} \cdot \vec{c} & \vec{c} \cdot \vec{c} \end{vmatrix} | \)
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