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