Twenty problems without step-by-step solutions, in three difficulty levels. Final answers at the end.
- ⭐ Easy — direct application
- ⭐⭐ Medium — multi-step, decomposition, combination
- ⭐⭐⭐ Hard — combines several sections
⭐ Problem 1 (§3.1)
Pick three vector quantities and three scalar quantities from: density, force, temperature, velocity, volume, acceleration, charge, momentum.
⭐ Problem 2 (§3.2)
Vector \vec a has length 5 and vector \vec b has length 3. Find the max and min possible |\vec a + \vec b|.
⭐ Problem 3 (§3.3)
\vec F = 100 N at 60° above horizontal. Compute F_x and F_y.
⭐ Problem 4 (§3.3)
Vector in component form: \vec r = 3\hat i - 4\hat j\ \mathrm{m}. Magnitude and angle with +x?
⭐⭐ Problem 5 (§3.4)
Three vectors: \vec a = 4\hat i + \hat j, \vec b = -2\hat i + 3\hat j, \vec c = \hat i - 5\hat j. Give \vec a + \vec b + \vec c in component form, magnitude, and angle.
⭐⭐ Problem 6 (§3.4)
An object moves from (2, 5) to (-3, -1). Displacement vector and its magnitude?
⭐⭐ Problem 7 (§3.5)
\vec a = 3\hat i + 4\hat j, \vec b = 5\hat i - 12\hat j. \vec a \cdot \vec b and angle between them?
⭐⭐ Problem 8 (§3.5)
\vec a = 2\hat i + k\hat j, \vec b = 3\hat i + \hat j. For what k are they perpendicular?
⭐⭐ Problem 9 (§3.5)
A car travels 500 m along a straight road while a horizontal 800 N force acts on it. Work done? What if the force is at 20° above the motion direction?
⭐⭐ Problem 10 (§3.6)
Write \hat i \times \hat j, \hat j \times \hat k, \hat k \times \hat i, \hat j \times \hat i without computation.
⭐⭐ Problem 11 (§3.6)
\vec a = \hat i + 2\hat j + 3\hat k, \vec b = 4\hat i + 5\hat j + 6\hat k. Compute \vec a \times \vec b.
⭐⭐ Problem 12 (§3.6)
A bolt at the origin. Force \vec F = 10\hat j + 5\hat k applied at position \vec r = 0.2\hat i. Torque?
⭐⭐⭐ Problem 13 (§3.2)
An airplane moves at 250 km/h relative to air, heading east. Wind blows 60 km/h toward southwest (225°). Speed relative to ground?
⭐⭐⭐ Problem 14 (§3.4)
Three forces on an object in equilibrium. If \vec F_1 = 30\hat i + 40\hat j and \vec F_2 = -50\hat i + 20\hat j, what is \vec F_3?
⭐⭐⭐ Problem 15 (§3.5)
Both \vec a and \vec b have magnitude 5. If |\vec a + \vec b| = 8, what is the angle between them?
⭐⭐⭐ Problem 16 (§3.6)
Area of a triangle with vertices A = (2, 0, 0), B = (0, 3, 0), C = (0, 0, 4)?
⭐⭐⭐ Problem 17 (§3.5 + §3.6)
Show that for any three vectors \vec a, \vec b, \vec c:
$$
(\vec a \times \vec b) \cdot \vec c = \vec a \cdot (\vec b \times \vec c)
$$
(the “scalar triple product” — a parallelepiped volume).
⭐⭐⭐ Problem 18 (§3.6)
A charged particle q = 1.6 \times 10^{-19} C moves at \vec v = 10^6\hat i\ \mathrm{m/s} in a magnetic field \vec B = 0.5\hat k\ \mathrm{T}. Lorentz force?
⭐⭐⭐ Problem 19 (§3.3)
Vector in spherical coordinates: magnitude 5, \theta = 60° from z, \phi = 30° in the xy plane. x, y, z components?
⭐⭐⭐ Problem 20 (challenge — §3.4 + §3.5)
If \vec a + \vec b = \vec c and \vec a \cdot \vec b = 0, prove c^2 = a^2 + b^2 (vector Pythagoras).
Final answers
1) Vectors: force, velocity, acceleration, momentum. Scalars: density, temperature, volume, charge.
2) Max: 8. Min: 2.
3) F_x = 50 N, F_y = 86.6 N.
4) r = 5 m, \theta = -53.1° (or 306.9°, quadrant IV).
5) 3\hat i - \hat j, magnitude \sqrt{10} \approx 3.16, angle -18.4°.
6) Displacement -5\hat i - 6\hat j, magnitude \sqrt{61} \approx 7.81 m.
7) \vec a \cdot \vec b = -33, \cos\phi = -0.508, \phi \approx 120.5°.
8) k = -6.
9) Horizontal: 400,000 J. At 20°: ~375,900 J.
10) \hat k, \hat i, \hat j, -\hat k.
11) \vec a \times \vec b = -3\hat i + 6\hat j - 3\hat k.
12) \vec \tau = -\hat j + 2\hat k\ \mathrm{N \cdot m}.
13) Total: 207.6\hat i - 42.4\hat j, magnitude 212 km/h, ~11.6° south of east.
14) \vec F_3 = 20\hat i - 60\hat j.
15) \cos\phi = 14/50 = 0.28, \phi \approx 73.7°.
16) \vec{AB} = -2\hat i + 3\hat j, \vec{AC} = -2\hat i + 4\hat k. Cross: 12\hat i + 8\hat j + 6\hat k, magnitude \sqrt{244} \approx 15.6. Area ~7.8.
17) Proof via component form — both sides equal a_x(b_y c_z - b_z c_y) + a_y(b_z c_x - b_x c_z) + a_z(b_x c_y - b_y c_x).
18) \vec F = q\vec v \times \vec B = -8 \times 10^{-14}\hat j\ \mathrm{N}.
19) x = 3.75, y = 2.17, z = 2.5.
20) c^2 = (\vec a + \vec b) \cdot (\vec a + \vec b) = a^2 + 2\vec a \cdot \vec b + b^2 = a^2 + b^2 (since \vec a \cdot \vec b = 0).
Preview of §3.9
Supplementary resources for deepening vectors — books, online courses, interactive tools.
📚 See also: Halliday Vol 1, Ch 3 — Problems.
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