- Identify the given vector \(\vec{v} = \langle 3, 4 \rangle\).
- Calculate the magnitude of the vector: \(||\vec{v}|| = \sqrt{3^2 + 4^2} = 5\).
- Divide each component by the magnitude to obtain the unit vector: \(\vec{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle\).
- Select the matching option: \(\boxed{\langle \frac{3}{5}, \frac{4}{5} \rangle}\).
Explanation
- Analysis of the Problem
The problem asks to find a unit vector in the same direction as the given vector \(\vec{v} = \langle 3, 4 \rangle\). A unit vector is a vector with a magnitude (length) of 1. For any non-zero vector \(\vec{v}\), a unit vector \(\vec{u}\) in the same direction can be found by dividing the vector \(\vec{v}\) by its magnitude \(||\vec{v}||\), such that \(\vec{u} = \frac{\vec{v}}{||\vec{v}||}\). The magnitude of a 2D vector \(\vec{v} = \langle x, y \rangle\) is defined as \(||\vec{v}|| = \sqrt{x^2 + y^2}\).
- Calculation of the Unit Vector
We first calculate the magnitude of the given vector \(\vec{v} = \langle 3, 4 \rangle\):
\[||\vec{v}|| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.\]
With the magnitude known, we find the unit vector \(\vec{u}\) by dividing each component of \(\vec{v}\) by the magnitude:
\[\vec{u} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle.\]
- Conclusion
Comparing our result \(\vec{u} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle\) with the given options, we find that the first option, \(\left\langle \frac{3}{5}, \frac{4}{5} \right\rangle\), is the correct unit vector in the same direction as \(\vec{v} = \langle 3, 4 \rangle\). Thus, the final answer is \(\boxed{\left\langle \frac{3}{5}, \frac{4}{5} \right\rangle}\).
Examples
In navigation and robotics, unit vectors are essential for representing directions independently of distance. For instance, if a drone is programmed to move in a specific direction (given by vector \(\langle 3, 4 \rangle\)), calculating the unit vector allows the controller to adjust the speed by simply scaling the unit vector by the desired speed value. This fundamental concept simplifies complex motion planning and trajectory control.