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

Transcript text: What are the magnitude and normalized vector, $\hat{v}=\frac{\vec{k}}{\|\vec{\nabla}\|}$, of $\vec{v}=(1,2,2) ?$

Solution

Step 1: Calculate the Magnitude of $\vec{v}$

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 $\vec{v}$:

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

Step 2: Normalize the Vector $\vec{v}$

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

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

The magnitude of $\vec{v}$ is:

\[ \boxed{\|\vec{v}\| = 3} \]

The normalized vector $\hat{v}$ is:

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

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