Let ` vec a , vec b , vec c` be three vectors such that `| vec a|=1,| vec b|=2a n d| vec c|=3.` If the projection of ` vec b` along a is equal to the projection of ` vec c` along ` vec aa n d vec b , vec c` are perpendicular to each other, find `|3 vec a-2 vec b+2 vec c|dot`

Explanation

To solve the problem step by step, we will follow the given conditions and apply the properties of vectors.

### Step 1: Understand the Given Information
We have three vectors:
- \( \vec{a} \) with magnitude \( |\vec{a}| = 1 \)
- \( \vec{b} \) with magnitude \( |\vec{b}| = 2 \)
- \( \vec{c} \) with magnitude \( |\vec{c}| = 3 \)

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