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Finding a Vector of Specific Magnitude in a Given Direction

To find a vector with a specific magnitude in the direction of a given vector, we first need to derive the unit vector in the direction of the given vector. Let’s break down the steps to find a vector of magnitude 7 in the direction of the vector v = 12i + 5k.

Step 1: Calculate the Magnitude of Vector v

The magnitude of vector v can be calculated using the formula:

||v|| = √(x² + y² + z²)

For our vector, which has components (12, 0, 5), the magnitude is:

||v|| = √(12² + 0² + 5²)
      = √(144 + 0 + 25)
      = √169 = 13

Step 2: Find the Unit Vector in the Direction of v

A unit vector in the direction of vector v is found by dividing each component of vector v by its magnitude:

u = (1/||v||) * v = (1/13) * (12i + 5k) = (12/13)i + (5/13)k

Step 3: Scale the Unit Vector to Desired Magnitude

To find a vector of magnitude 7 in the direction of unit vector u, we multiply it by 7:

w = 7 * u = 7 * [(12/13)i + (5/13)k] = (84/13)i + (35/13)k

Final Result

Thus, the vector of magnitude 7 in the direction of 12i + 5k is:

w = (84/13)i + (35/13)k

This vector maintains the same direction as the original vector while having the specified magnitude of 7.

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