Two forces act on an object: 6 N pointing east and 8 N pointing north. What is the magnitude of their resultant?
A14 N, since 6 + 8 = 14
B10 N, found by applying the Pythagorean theorem to the perpendicular components
C2 N, since the forces partially cancel
D48 N, since forces multiply when combined
Because the forces are perpendicular, |u + v| = √(6² + 8²) = √100 = 10 N. Adding magnitudes directly (giving 14) is the classic error — magnitudes add linearly only when both vectors point in exactly the same direction. For perpendicular vectors, the Pythagorean theorem applies. The correct approach is always to add component-wise first, then compute the magnitude of the resulting vector.
Question 2 Multiple Choice
Point P is at position (3, 5) and point Q is at position (7, 2). Which vector represents the displacement from P to Q?
A(10, 7), by adding the position vectors of P and Q
B(4, −3), by computing Q − P
C(−4, 3), by computing P − Q
D(3, 5), since displacement is measured from the origin
The displacement from P to Q is Q − P = (7−3, 2−5) = (4, −3). This follows from the key geometric interpretation: u − v is the vector *from* v *to* u. So to go from P to Q, compute Q − P. Option C, P − Q = (−4, 3), is the displacement from Q back to P — the opposite direction. Adding position vectors (option A) produces (10, 7), which has no geometric meaning as a displacement between specific points.
Question 3 True / False
If vector u has magnitude 5 and vector v has magnitude 5, then u + v is expected to have magnitude 10.
TTrue
FFalse
Answer: False
Magnitudes do not add in general. |u + v| = |u| + |v| only when u and v point in exactly the same direction. If they point in opposite directions, |u + v| = 0. If they are perpendicular, |u + v| = √50 ≈ 7.07. The magnitude of a sum must be computed from the sum vector itself — not from the individual magnitudes — using the Pythagorean theorem or the full component calculation. The triangle inequality |u + v| ≤ |u| + |v| shows that 10 is the maximum, achieved only in the collinear case.
Question 4 True / False
The vector u − v can be interpreted geometrically as the vector pointing from the tip of v to the tip of u, when both vectors are drawn from the origin.
TTrue
FFalse
Answer: True
This is the key geometric interpretation of subtraction. If u and v are position vectors of points P and Q respectively, then u − v is the displacement from Q (tip of v) to P (tip of u). Equivalently, u − v = u + (−v): negate v (reverse its direction), then add tip-to-tail. Getting the direction right is essential in applications like computing displacement paths between two points, normal vectors in geometry, and relative velocity in physics.
Question 5 Short Answer
Explain why the magnitude of u + v is not generally equal to |u| + |v|, and describe when equality does hold.
Think about your answer, then reveal below.
Model answer: Vector addition is component-wise: (u₁ + v₁, u₂ + v₂). The magnitude of the sum is √((u₁+v₁)² + (u₂+v₂)²), which does not generally simplify to √(u₁²+u₂²) + √(v₁²+v₂²). Equality holds only when u and v point in exactly the same direction — in that case the components scale proportionally and the Pythagorean theorem reduces to simple addition. In all other cases the angle between the vectors reduces the magnitude of the sum, which is precisely what the triangle inequality captures: |u + v| ≤ |u| + |v| with equality only in the collinear, same-direction case.
A concrete example: u = (3, 0), v = (0, 4). |u| = 3, |v| = 4, sum of magnitudes = 7. But u + v = (3, 4), and |u + v| = √(9+16) = 5. The same numbers as the classic 3-4-5 right triangle — because that's exactly what's happening geometrically. The vectors are perpendicular, so the Pythagorean theorem gives the magnitude of the sum.