Questions: What are the magnitude and normalized vector, hat(v) = vec(k) / vec(∇), of vec(v) = (1,2,2)?

Solution Steps

To find the magnitude of the vector \(\vec{v} = (1, 2, 2)\), we use the formula for the Euclidean norm:

\[ \|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]

Substituting the components of \(\vec{v}\):

\[ \|\vec{v}\| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

To find the normalized vector \(\hat{v}\), we divide each component of \(\vec{v}\) by its magnitude \(\|\vec{v}\|\):

\[ \hat{v} = \frac{\vec{v}}{\|\vec{v}\|} = \left( \frac{1}{3}, \frac{2}{3}, \frac{2}{3} \right) \]

Final Answer