How to find a unit vector in the same direction?

Understand the Problem

The question is asking us to explain the method for finding a unit vector that points in the same direction as a given vector. To solve this, we need to discuss the process of normalizing a vector, which involves dividing the vector by its magnitude.

Answer

The unit vector is given by $\hat{v} = \left\langle \frac{v_1}{|\vec{v}|}, \frac{v_2}{|\vec{v}|}, \frac{v_3}{|\vec{v}|} \right\rangle$.

Steps to Solve

Identify the given vector

Start by noting the components of the vector you want to normalize. Let the vector be represented as $\vec{v} = \langle v_1, v_2, v_3 \rangle$ for a 3-dimensional vector.

Calculate the magnitude of the vector

The magnitude (or length) of the vector is calculated using the formula:

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

Normalize the vector

To find the unit vector, divide each component of the vector by its magnitude. The unit vector $\hat{v}$ is given by:

$$ \hat{v} = \left\langle \frac{v_1}{|\vec{v}|}, \frac{v_2}{|\vec{v}|}, \frac{v_3}{|\vec{v}|} \right\rangle $$

Resulting unit vector

The resulting vector $\hat{v}$ is the unit vector that points in the same direction as the original vector $\vec{v}$.

The unit vector $\hat{v}$ is given by:

$$ \hat{v} = \left\langle \frac{v_1}{|\vec{v}|}, \frac{v_2}{|\vec{v}|}, \frac{v_3}{|\vec{v}|} \right\rangle $$

More Information

A unit vector has a magnitude of 1 and represents direction only, making it essential in various applications within physics and engineering, such as defining directions in space.

Tips

Forgetting to compute the magnitude correctly can lead to incorrect results when creating the unit vector.
Dividing only one component of the vector by the magnitude, instead of all components, is a common error.

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