Questions: Vectors in 3D Space: Operations and Magnitude

Question 1

What is the magnitude of the vector ⟨-3, 0, 4⟩?

• A. 7
• B. 5
• C. -5
• D. 1

Answer: (B)

Magnitude = √((-3)² + 0² + 4²) = √(9 + 0 + 16) = √25 = 5. The negative sign on the component is eliminated by squaring — magnitude is always non-negative. A common error is adding the components directly (−3 + 0 + 4 = 1) instead of squaring them.

Question 2

Multiplying a vector by a negative scalar reverses its direction while keeping its magnitude unchanged.

• T. True
• F. False

Answer: False

Only multiplication by −1 preserves magnitude while reversing direction. Multiplying by any other negative scalar −k both reverses the direction and scales the magnitude by |k|. For example, −3 × ⟨1, 0, 0⟩ = ⟨−3, 0, 0⟩, which has magnitude 3, not 1.

Question 3

Why is 3D vector addition computed component-wise rather than some other way?

Model answer: Because the three coordinate axes are mutually perpendicular (orthogonal), displacement along each axis is independent. Adding component-wise accumulates displacement along each independent direction separately, which corresponds geometrically to the tip-to-tail rule.

When two axes are orthogonal, motion along one does not affect position along the other. So total x-displacement, y-displacement, and z-displacement each add independently — exactly what component-wise addition computes. Geometrically, this is equivalent to placing vectors end-to-end and measuring the resultant displacement.