The left side = |A•i|² + |A•j|² + |A•k|² = |A|²
The right side = |A|²

Let any vector A = < p,q,r >

|A•i|² = |< p,q,r >•< 1,0,0 >|² = |(p)(1)+(q)(0)+(r)(0)|² = |p|² = p²

|A•j|² = |< p,q,r >•< 0,1,0 >|² = |(p)(0)+(q)(1)+(r)(0)|² = |q|² = q²

|A•k|² = |< p,q,r >•< 0,0,1 >|² = |(p)(0)+(q)(0)+(r)(1)|² = |r|² = r²

So the left side = |A•i|² + |A•j|² + |A*k|² = p² + q² + r²

The right side = |A|² = |√p² + q² + r²|² = p² + q² + r²

Edwin