How can I find a vector of magnitude 3 that points in the same direction as the vector represented by 10i + 24k?

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To find a vector of a specific magnitude that points in the same direction as a given vector, we can follow these steps:

  1. Identify the original vector: The vector given is v = 10i + 24k.
  2. Calculate the magnitude of the original vector: The magnitude (length) of a vector v = ai + bj + ck is calculated using the formula:

    3. \[ ||v|| = \sqrt{a^2 + b^2 + c^2} \]

    For our vector:

  3. Here, a = 10, b = 0, and c = 24. Substituting these values into the formula:

    \[ ||v|| = \sqrt{10^2 + 0^2 + 24^2} = \sqrt{100 + 0 + 576} = \sqrt{676} = 26 \]

  4. Find the unit vector in the direction of v: The unit vector u in the direction of v is calculated by dividing v by its magnitude:

    5. \[ u = \frac{v}{||v||} = \frac{10i + 24k}{26} \]

    Which simplifies to:

    \[ u = \left( \frac{10}{26}i + \frac{24}{26}k \right) = \left( \frac{5}{13}i + \frac{12}{13}k \right) \]

  5. Scale the unit vector to the desired magnitude: To find a vector of magnitude 3 in the same direction, we multiply the unit vector u by 3:

    6. \[ v_{desired} = 3u = 3 \left( \frac{5}{13}i + \frac{12}{13}k \right) \]

    Thus,

    \[ v_{desired} = \left( \frac{15}{13}i + \frac{36}{13}k \right) \]

  6. Final Result: Therefore, a vector of magnitude 3 in the direction of 10i + 24k is:

    7. \[ \left( \frac{15}{13}i + \frac{36}{13}k \right) \]

With this process, you can find any vector with a specified magnitude in the direction of a given vector!

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