Find the length of the vector \(v = (4, 3, 12)\).

Step 1 ：We are given the vector \( v = (4, 3, 12) \). The length (or magnitude) of a vector \(v = (a, b, c)\) in three-dimensional space is given by the formula \( ||v|| = \sqrt{a^2 + b^2 + c^2} \).

Step 2 ：Substitute the values into the formula: \( ||v|| = \sqrt{4^2 + 3^2 + 12^2} \).

Step 3 ：Solve the equation: \( ||v|| = \sqrt{16 + 9 + 144} \).

Step 4 ：Simplify the equation: \( ||v|| = \sqrt{169} \).