Let `vecp and vecq` any two othogonal vectors of equal magnitude 4 each. Let `veca ,vecb and vecc` be any three vectors of lengths `7sqrt15 and 2sqrt33`, mutually perpendicular to each other. Then find the distance of the vector `(veca.vecp)vecp+(veca.vecq)vecq+(veca.(vecpxxvecq))(vecpxxvecq)+(vecb.vecp)vecp+(vecb.vecp)vecq+ `
`(vecb.(vecb.vecq))(vecpxxvecq)+(vecc.vecp)vecp+(vecc.vecq)vecq+(vecc.(vecpxxvecq))(vecpxxvecq)` from the origin.

Explanation

To solve the problem, we need to find the distance of the vector

\[ \vec{a} \cdot \vec{p} \vec{p} + \vec{a} \cdot \vec{q} \vec{q} + \vec{a} \cdot (\vec{p} \times \vec{q})(\vec{p} \times \vec{q}) + \vec{b} \cdot \vec{p} \vec{p} + \vec{b} \cdot \vec{q} \vec{q} + \vec{b} \cdot (\vec{b} \cdot \vec{q})(\vec{p} \times \vec{q}) + \vec{c} \cdot \vec{p} \vec{p} + \vec{c} \cdot \vec{q} \vec{q} + \vec{c} \cdot (\vec{p} \times \vec{q})(\vec{p} \times \vec{q}) \]

from the origin.

### Step 1: Understanding the Given Vectors
Given:
- \(\vec{p}\) and \(\vec{q}\) are orthogonal vectors of equal magnitude \(4\).
- \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are mutually perpendicular vectors with lengths \(7\sqrt{15}\) and \(2\sqrt{33}\).

### Step 2: Calculate the Components
Since \(\vec{p}\) and \(\vec{q}\) are orthogonal, we can denote:
- \(|\vec{p}| = |\vec{q}| = 4\)
- \(|\vec{p} \times \vec{q}| = |\vec{p}| |\vec{q}| = 4 \cdot 4 = 16\)

### Step 3: Evaluate Each Term
1. **For \(\vec{a}\)**:
\[ \vec{a} \cdot \vec{p} \vec{p} + \vec{a} \cdot \vec{q} \vec{q} + \vec{a} \cdot (\vec{p} \times \vec{q})(\vec{p} \times \vec{q}) \]
This simplifies to:
\[ (\vec{a} \cdot \vec{p}) \vec{p} + (\vec{a} \cdot \vec{q}) \vec{q} + (\vec{a} \cdot (\vec{p} \times \vec{q})) (\vec{p} \times \vec{q}) \]
Since \(\vec{a}\) is perpendicular to \(\vec{p}\) and \(\vec{q}\), the dot products \(\vec{a} \cdot \vec{p}\) and \(\vec{a} \cdot \vec{q}\) will be zero. Thus, we only need to consider the last term.

2. **For \(\vec{b}\)**:
Similarly, we can evaluate:
\[ \vec{b} \cdot \vec{p} \vec{p} + \vec{b} \cdot \vec{q} \vec{q} + \vec{b} \cdot (\vec{b} \cdot \vec{q})(\vec{p} \times \vec{q}) \]
Again, the first two terms will be zero, leaving us with the last term.

3. **For \(\vec{c}\)**:
The same logic applies:
\[ \vec{c} \cdot \vec{p} \vec{p} + \vec{c} \cdot \vec{q} \vec{q} + \vec{c} \cdot (\vec{p} \times \vec{q})(\vec{p} \times \vec{q}) \]
Again, the first two terms will be zero.

### Step 4: Combine the Results
The total vector can be simplified to:
\[ (\vec{a} \cdot (\vec{p} \times \vec{q})) (\vec{p} \times \vec{q}) + (\vec{b} \cdot (\vec{p} \times \vec{q})) (\vec{p} \times \vec{q}) + (\vec{c} \cdot (\vec{p} \times \vec{q})) (\vec{p} \times \vec{q}) \]
This can be factored out:
\[ (\vec{a} \cdot (\vec{p} \times \vec{q}) + \vec{b} \cdot (\vec{p} \times \vec{q}) + \vec{c} \cdot (\vec{p} \times \vec{q})) (\vec{p} \times \vec{q}) \]

### Step 5: Calculate the Magnitude
The distance from the origin is given by the magnitude of the resulting vector:
\[ \sqrt{(\vec{a}^2 + \vec{b}^2 + \vec{c}^2) \cdot |\vec{p}|^2} \]

Substituting the values:
- \(|\vec{p}|^2 = 4^2 = 16\)
- \(|\vec{a}|^2 = (7\sqrt{15})^2 = 735\)
- \(|\vec{b}|^2 = (2\sqrt{33})^2 = 132\)

Thus,
\[ \text{Distance} = \sqrt{735 + 132} \cdot 4 = \sqrt{867} \cdot 4 = 4\sqrt{867} \]

### Final Answer
The distance from the origin is \(4\sqrt{867}\).