To find the unit vector in the direction of the vector $\langle -2, 5 \rangle$, follow these steps:

Step 1: Find the magnitude of the vector

The magnitude of the vector $\langle -2, 5 \rangle$ is calculated using the formula: $\| \langle -2, 5 \rangle \| = \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}$

Step 2: Divide each component by the magnitude

To obtain the unit vector, divide each component of the original vector by its magnitude: $\text{Unit vector} = \left\langle \frac{-2}{\sqrt{29}}, \frac{5}{\sqrt{29}} \right\rangle$

Final Answer

The unit vector in the direction of $\langle -2, 5 \rangle$ is: $\left\langle \frac{-2}{\sqrt{29}}, \frac{5}{\sqrt{29}} \right\rangle$

Would you like further details or clarification?

Here are some related questions:

1. How do we calculate the unit vector for a 3D vector?
2. Why is the unit vector important in vector calculations?
3. What does the direction of the unit vector represent in a coordinate system?
4. How does this unit vector change if we reverse the original vector?
5. Can the magnitude of any unit vector be different from 1?

Tip: A unit vector always has a magnitude of 1, making it a useful tool for indicating direction without altering the vector’s length.