Find the magnitude of the vector v = (3, -4, 12).

Step 1: Recall the magnitude formula.

For a vector v = (a, b, c), its magnitude is:

||v|| = √(a^2 + b^2 + c^2)

Step 2: Substitute the components.

||v|| = √(3^2 + (-4)^2 + 12^2)

Step 3: Evaluate the squares.

3^2 = 9
(-4)^2 = 16
12^2 = 144

Therefore:
||v|| = √(9 + 16 + 144)

Step 4: Add inside the square root.

9 + 16 + 144 = 169

So:
||v|| = √169

Step 5: Take the square root.

||v|| = 13

Therefore, the magnitude of v is 13.