Find the length of the vector \( \vec{v} = 3\hat{i} - 2\hat{j} + 4\hat{k} \)

Step 1 ：The length or magnitude of a vector \(\vec{v}\) is given by \(||\vec{v}|| = \sqrt{x^2 + y^2 + z^2}\), where \(x, y, z\) are the components of the vector.

Step 2 ：Substitute the components of the vector \(\vec{v}\) into the formula: \(||\vec{v}|| = \sqrt{(3)^2 + (-2)^2 + (4)^2}\)

Step 3 ：Simplify the expression: \(||\vec{v}|| = \sqrt{9 + 4 + 16}\)

Step 4 ：Calculate the square root: \(||\vec{v}|| = \sqrt{29}\)