Questions: Scalar and Vector Mechanics

Question 1

A 5 N force points due east and a second 5 N force points due north. Both are applied to the same object. What is the magnitude of the resultant force?

• A. 10 N, because 5 + 5 = 10
• B. 0 N, because the forces are perpendicular and cancel out
• C. 5√2 ≈ 7.07 N, because perpendicular vectors combine by the Pythagorean theorem
• D. 5 N, because the magnitudes are equal so only one counts

Answer: (C)

This is the canonical example of why scalar addition fails for vectors in different directions. The resultant of two perpendicular equal forces is found by the Pythagorean theorem: √(5² + 5²) = √50 = 5√2 ≈ 7.07 N, pointing northeast at 45°. Adding magnitudes (5 + 5 = 10 N) would only be correct if both forces pointed in exactly the same direction. Perpendicular forces do not cancel — that would require equal and opposite forces, i.e., same line, opposite directions.

Question 2

Which of the following pairs correctly classifies one quantity as a scalar and one as a vector?

• A. Velocity (scalar) and speed (vector)
• B. Work (scalar) and force (vector)
• C. Displacement (scalar) and distance (vector)
• D. Temperature (vector) and pressure (scalar)

Answer: (B)

Work = F · d (dot product of force and displacement) produces a scalar — it has magnitude but no direction. Force requires both magnitude and direction — it is a vector. The other options are reversed: velocity is a vector (has direction), speed is a scalar (magnitude only); displacement is a vector, distance is a scalar; temperature and pressure are both scalars. The dot product is precisely the operation that 'collapses' two vectors into a scalar by extracting their aligned component.

Question 3

If two forces of equal magnitude act on an object in different directions, they typically cancel out and the net force is zero.

• T. True
• F. False

Answer: False

Forces cancel only when they are equal in magnitude AND opposite in direction — a very specific relationship. Two equal forces in different directions (e.g., east and north) combine to produce a nonzero resultant. Even forces at 179° to each other produce a small nonzero net force pointing in the direction of their bisector. Cancellation is the special case of exactly 180° between the force vectors. This confusion often arises from treating forces as scalars and subtracting magnitudes, when the full vector addition is required.

Question 4

The magnitude of velocity and the magnitude of speed are equal when an object moves along a straight line in one direction.

• T. True
• F. False

Answer: True

Speed is the scalar magnitude of the velocity vector — the rate of change of distance. Velocity is the rate of change of displacement, a vector quantity. When an object moves in a straight line without reversing direction, the path length equals the magnitude of the displacement, so speed and the magnitude of velocity are equal. They diverge when the path curves or reverses: an object that travels 5 m east and then 5 m west has covered 10 m (speed-based distance) but has zero net displacement — average velocity is 0, average speed is not.

Question 5

Two students are adding the forces on a bridge joint: a 3 kN force pointing 60° above horizontal and a 4 kN force pointing straight down. One student adds the magnitudes to get 7 kN. Explain why this is wrong and how to find the correct resultant.

Model answer: Adding magnitudes is only valid when forces point in the same direction. These forces are in different directions, so components must be added separately. Decompose each: the 3 kN force has components (3cos60° = 1.5 kN horizontal, 3sin60° ≈ 2.6 kN upward); the 4 kN force is (0 horizontal, −4 kN vertical). Sum: Fx = 1.5 kN, Fy = 2.6 − 4 = −1.4 kN. Resultant magnitude = √(1.5² + 1.4²) ≈ 2.05 kN — far less than 7 kN.

Scalar addition of vector magnitudes is a category error. It gives the right answer only in the degenerate case of parallel, same-direction forces. The component method — decompose all forces into x and y (and z in 3D), add each axis separately, then reconstruct the resultant — is the universal procedure. The 7 kN answer would imply the joint experiences more force than if both forces were aligned, which violates physical intuition: cancellation of opposing components always reduces the resultant below the sum of magnitudes.