How can I find a unit vector that shares the same direction as the vector 5i + 9j?

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To find a unit vector that has the same direction as the vector 5i + 9j, we first need to understand what a unit vector is. A unit vector is a vector that has a length (or magnitude) of 1, but points in the same direction as the original vector.

Here’s how you can find it step-by-step:

  1. Calculate the magnitude of the vector: The magnitude (or length) of the vector 5i + 9j can be calculated using the formula:

    2. |v| = √(x2 + y2)

    In our case, x = 5 and y = 9, so:

    |v| = √(52 + 92) = √(25 + 81) = √106

  2. Divide the vector by its magnitude: To convert the vector into a unit vector, we divide each component of the vector by its magnitude:

    3. The unit vector u in the same direction as 5i + 9j is given by:

    u = (1/|v|) * (5i + 9j)

    This can be rewritten as:

    u = (5/√106)i + (9/√106)j

  3. Final result: The unit vector in the direction of 5i + 9j is:

    4. u = (5/√106)i + (9/√106)j

This unit vector maintains the same direction as your original vector while having a length of 1. It can be useful in a variety of applications, such as normalizing vectors in physics and computer graphics.

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