How do I find the unit vector that has the same direction as the vector v = 3i + j?

To find the unit vector in the same direction as the vector v = 3i + j, we follow a straightforward process that involves calculating the magnitude of the vector and then dividing each component of the vector by its magnitude.

Here’s a step-by-step breakdown:

1. Identify the components: The vector v = 3i + j has components (3, 1), where 3 is the coefficient of i (the x-component) and 1 is the coefficient of j (the y-component).
2. Calculate the magnitude: The magnitude (or length) of the vector is given by the formula:

||v|| = √(x² + y²)

For our vector:

||v|| = √(3² + 1²) = √(9 + 1) = √(10)

3. Divide each component by the magnitude: Now that we have the magnitude, we can find the unit vector u in the same direction as v:

u = (1/||v||) * v

u = (1/√{10}) * (3i + j) = (3/√{10})i + (1/√{10})j

Thus, the unit vector in the same direction as v = 3i + j is:

u = (3/√{10})i + (1/√{10})j

Alternatively, you can express the unit vector in decimal form if you prefer:

u ≈ 0.9487i + 0.3162j

In summary, the unit vector that has the same direction as the vector v = 3i + j is approximately:

u ≈ 0.9487i + 0.3162j