Which expression verifies that a unit vector u = (a,b,c) has magnitude 1?

• A. |u| = sqrt(a^2+b^2+c^2)
• B. |u| = a^2+b^2+c^2
• C. |u| = a+b+c
• D. |u| = 1

Answer: (A)

The key idea is how we measure the length of a vector in three dimensions. The length (magnitude) of a vector (a, b, c) is given by the Euclidean norm, which strips out the square roots of the sums of squares of its components: |u| = sqrt(a^2 + b^2 + c^2).

For a unit vector, this length must be 1, so we would have sqrt(a^2 + b^2 + c^2) = 1, meaning a^2 + b^2 + c^2 = 1. The expression that directly gives the magnitude in terms of a, b, and c is sqrt(a^2 + b^2 + c^2), making it the correct form to verify unit length.

The other forms don’t represent the magnitude in terms of the components: without the square root, you get the squared length; summing the components a + b + c isn’t related to length in general; and simply stating |u| = 1 does describe the unit length but doesn’t express the magnitude in terms of a, b, and c.