To find a unit vector that has the same direction as the vector 3i + 7j, we first need to determine the magnitude of the given vector. The magnitude of a vector v = ai + bj is calculated using the formula:
|v| = √(a² + b²)
In this case, a = 3 and b = 7. Plugging in these values, we get:
|v| = √(3² + 7²) = √(9 + 49) = √58
Now that we have the magnitude, we can find the unit vector by dividing the original vector by its magnitude. The formula for a unit vector u in the direction of v is:
u = (1/|v|) * v
Substituting the values, we have:
u = (1/√58) * (3i + 7j)
Thus, the unit vector that points in the same direction as 3i + 7j is:
u = (3/√58)i + (7/√58)j
This unit vector maintains the direction of the original vector while having a magnitude of 1.