To find a unit vector that has the same direction as a given vector, you need to first understand what a unit vector is. A unit vector is a vector with a magnitude (length) of one. The process of finding a unit vector involves two key steps: determining the magnitude of the original vector and then dividing each component of the vector by this magnitude.
In this case, the given vector is:
v = 5i + 7j
1. **Calculate the Magnitude of the Vector**:
The magnitude of vector v can be calculated using the formula:
|v| = √(x2 + y2), where x and y are the components of the vector.
For our vector:
- x = 5
- y = 7
Substituting the values:
|v| = √(52 + 72) = √(25 + 49) = √74
2. **Divide Each Component by the Magnitude**:
To find the unit vector, u, in the same direction as v, simply divide each component of v by its magnitude:
u = (1/|v|) * v = (1/√74) * (5i + 7j)
This gives:
- u = (5/√74)i + (7/√74)j
3. **Result**:
Therefore, the unit vector in the same direction as the vector 5i + 7j is:
u = (5/√74)i + (7/√74)j
This unit vector retains the direction of the original vector but has a length of 1, making it ideal for various applications in physics, engineering, and computer graphics.