Problem Set 1: Vectors#
The problems below are intended to be done by hand using pencil/pen and paper.
Section A: Basic Vector Operations#
Problem 1. Given vectors \(\mathbf{a} = \begin{bmatrix} 3 \\ -2 \\ 5 \end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix}\), find \(\mathbf{a} + \mathbf{b}\).
Problem 2. If \(\mathbf{u} = \begin{bmatrix} 7 \\ -3 \end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix} 2 \\ 8 \end{bmatrix}\), calculate \(\mathbf{u} - \mathbf{v}\) and sketch \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{u} - \mathbf{v}\) in the Cartesian plane.
Problem 3. For \(\mathbf{p} = \begin{bmatrix} 4 \\ -1 \\ 6 \end{bmatrix}\), find \(3\mathbf{p}\).
Problem 4. Given \(\mathbf{x} = \begin{bmatrix} 2 \\ 5 \\ -3 \end{bmatrix}\) and \(\mathbf{y} = \begin{bmatrix} -4 \\ 1 \\ 7 \end{bmatrix}\), compute \(2\mathbf{x} - 3\mathbf{y}\).
Problem 5. If \(\mathbf{a} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}\), and \(\mathbf{c} = \begin{bmatrix} -2 \\ 4 \end{bmatrix}\), find \(\mathbf{a} + \mathbf{b} + \mathbf{c}\).
Problem 6. Calculate \(-5\mathbf{w}\) where \(\mathbf{w} = \begin{bmatrix} 2 \\ -3 \\ 1 \\ 4 \end{bmatrix}\).
Problem 7. For vectors \(\mathbf{m} = \begin{bmatrix} 6 \\ -2 \\ 8 \end{bmatrix}\) and \(\mathbf{n} = \begin{bmatrix} 3 \\ -1 \\ 4 \end{bmatrix}\), verify that \(\mathbf{m} = 2\mathbf{n}\).
Problem 8. Given \(\mathbf{r} = \begin{bmatrix} -3 \\ 7 \\ 2 \end{bmatrix}\) and \(\mathbf{s} = \begin{bmatrix} 5 \\ -2 \\ -1 \end{bmatrix}\), find \(\frac{1}{2}(\mathbf{r} + \mathbf{s})\).
Problem 9. If \(\mathbf{d} = \begin{bmatrix} 4 \\ -6 \\ 8 \end{bmatrix}\), find a vector \(\mathbf{e}\) such that \(\mathbf{d} + \mathbf{e} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\).
Problem 10. Calculate \(4\mathbf{a} - 2\mathbf{b} + \mathbf{c}\) where \(\mathbf{a} = \begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix} 3 \\ 4 \\ 1 \end{bmatrix}\), and \(\mathbf{c} = \begin{bmatrix} -1 \\ 2 \\ 5 \end{bmatrix}\).
Section B: Vector Norms and Magnitudes#
Problem 11. Find \(\|\mathbf{v}\|\) where \(\mathbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\).
Problem 12. Calculate the magnitude of \(\mathbf{w} = \begin{bmatrix} -2 \\ 1 \\ 2 \end{bmatrix}\).
Problem 13. If \(\mathbf{u} = \begin{bmatrix} 1 \\ -2 \\ 3 \\ -4 \end{bmatrix}\), find \(\|\mathbf{u}\|\).
Problem 14. Determine \(\|\mathbf{a}\|\) where \(\mathbf{a} = \begin{bmatrix} 5 \\ 0 \\ -12 \end{bmatrix}\).
Problem 15. Given \(\mathbf{p} = \begin{bmatrix} -6 \\ 8 \end{bmatrix}\), calculate \(\|\mathbf{p}\|^2\).
Problem 16. Find the magnitude of \(\mathbf{q} = \begin{bmatrix} 2 \\ -3 \\ 6 \\ 1 \end{bmatrix}\).
Problem 17. If \(\mathbf{r} = \begin{bmatrix} a \\ 2a \\ -a \end{bmatrix}\) where \(a > 0\), express \(\|\mathbf{r}\|\) in terms of \(a\).
Problem 18. Calculate \(\|3\mathbf{v}\|\) where \(\mathbf{v} = \begin{bmatrix} 2 \\ -1 \\ 4 \end{bmatrix}\).
Problem 19. Given \(\mathbf{x} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}\), find \(\|\mathbf{x}\|\).
Problem 20. If \(\|\mathbf{u}\| = 7\) and \(\mathbf{u} = \begin{bmatrix} 3 \\ -4 \\ b \end{bmatrix}\), find the value of \(b\).
Section C: Triangle Inequality#
Problem 21. For \(\mathbf{a} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix} 1 \\ -4 \end{bmatrix}\), verify the triangle inequality: \(\|\mathbf{a} + \mathbf{b}\| \leq \|\mathbf{a}\| + \|\mathbf{b}\|\).
Problem 22. Given \(\mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix} -2 \\ 1 \\ 3 \end{bmatrix}\), check if \(\|\mathbf{u} - \mathbf{v}\| \geq |\|\mathbf{u}\| - \|\mathbf{v}\||\) holds.
Problem 23. For vectors \(\mathbf{p} = \begin{bmatrix} 4 \\ 0 \end{bmatrix}\) and \(\mathbf{q} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\), calculate both sides of the triangle inequality and verify it holds.
Problem 24. If \(\mathbf{x} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) and \(\mathbf{y} = \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}\), show that \(\|\mathbf{x} + \mathbf{y}\| \leq \|\mathbf{x}\| + \|\mathbf{y}\|\).
Problem 25. Given \(\mathbf{m} = \begin{bmatrix} -3 \\ 4 \\ 0 \end{bmatrix}\) and \(\mathbf{n} = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}\), verify the reverse triangle inequality: \(\|\mathbf{m} - \mathbf{n}\| \geq |\|\mathbf{m}\| - \|\mathbf{n}\||\).
Problem 26. For \(\mathbf{a} = \begin{bmatrix} 5 \\ -2 \\ 1 \end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix} -1 \\ 3 \\ 2 \end{bmatrix}\), and \(\mathbf{c} = \begin{bmatrix} 2 \\ 1 \\ -3 \end{bmatrix}\), verify that \(\|\mathbf{a} + \mathbf{b} + \mathbf{c}\| \leq \|\mathbf{a}\| + \|\mathbf{b}\| + \|\mathbf{c}\|\).
Problem 27. If \(\mathbf{u}\) and \(\mathbf{v}\) are vectors such that \(\|\mathbf{u}\| = 5\) and \(\|\mathbf{v}\| = 3\), what are the possible range of values for \(\|\mathbf{u} + \mathbf{v}\|\)?
Problem 28. Given that \(\|\mathbf{r}\| = 4\) and \(\|\mathbf{s}\| = 7\), find the range of possible values for \(\|\mathbf{r} - \mathbf{s}\|\).
Section D: Unit Vectors#
Problem 29. Find the unit vector in the direction of \(\mathbf{v} = \begin{bmatrix} 6 \\ -8 \end{bmatrix}\).
Problem 30. Given \(\mathbf{w} = \begin{bmatrix} 2 \\ 1 \\ -2 \end{bmatrix}\), find the unit vector \(\hat{\mathbf{w}}\).
Problem 31. If \(\mathbf{u} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix}\), calculate the unit vector parallel to \(\mathbf{u}\).
Problem 32. Find a unit vector in the direction of \(\mathbf{a} = \begin{bmatrix} 1 \\ -1 \\ 1 \\ -1 \end{bmatrix}\).
Problem 33. Given \(\mathbf{p} = \begin{bmatrix} 0 \\ 5 \\ 0 \end{bmatrix}\), find the unit vector in the direction of \(\mathbf{p}\).
Problem 34. If \(\mathbf{r} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}\), find the unit vector \(\hat{\mathbf{r}}\) and verify that \(\|\hat{\mathbf{r}}\| = 1\).
Problem 35. Find two unit vectors that are parallel to \(\mathbf{v} = \begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}\).
Problem 36. Given \(\mathbf{q} = \begin{bmatrix} a \\ 2a \\ 2a \end{bmatrix}\) where \(a \neq 0\), find the unit vector in the direction of \(\mathbf{q}\).
Problem 37. If \(\hat{\mathbf{u}}\) is a unit vector in the direction of \(\mathbf{u} = \begin{bmatrix} 3 \\ -4 \\ 12 \end{bmatrix}\), find \(5\hat{\mathbf{u}}\).
Problem 38. Find the unit vector that points in the opposite direction of \(\mathbf{w} = \begin{bmatrix} 1 \\ 2 \\ -3 \end{bmatrix}\).
Section E: Mixed Problems#
Problem 39. Given \(\mathbf{a} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix} -1 \\ 4 \\ 1 \end{bmatrix}\), find \(\|\mathbf{a} + \mathbf{b}\|\) and compare it to \(\|\mathbf{a}\| + \|\mathbf{b}\|\).
Problem 40. If \(\mathbf{u} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}\) and \(\hat{\mathbf{v}}\) is a unit vector in the direction of \(\mathbf{v} = \begin{bmatrix} 5 \\ 12 \end{bmatrix}\), find \(\mathbf{u} + 2\hat{\mathbf{v}}\).
Problem 41. For \(\mathbf{p} = \begin{bmatrix} 1 \\ 0 \\ -2 \\ 3 \end{bmatrix}\), find scalars \(c_1\) and \(c_2\) such that \(c_1\mathbf{p} + c_2(-\mathbf{p})\) is a unit vector.
Problem 42. Given vectors \(\mathbf{x} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\) and \(\mathbf{y} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}\), find unit vectors \(\hat{\mathbf{x}}\) and \(\hat{\mathbf{y}}\), then calculate \(\|\hat{\mathbf{x}} + \hat{\mathbf{y}}\|\).
Problem 43. If \(\mathbf{a} = \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}\), find the vector \(\mathbf{b}\) such that \(\mathbf{b}\) has the same direction as \(\mathbf{a}\) but has magnitude 10.
Problem 44. For \(\mathbf{r} = \begin{bmatrix} -6 \\ 8 \end{bmatrix}\) and \(\mathbf{s} = \begin{bmatrix} 3 \\ -4 \end{bmatrix}\), find the angle between the unit vectors \(\hat{\mathbf{r}}\) and \(\hat{\mathbf{s}}\).
Problem 45. Given \(\mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}\), find a unit vector perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\).
Problem 46. If \(\|\mathbf{a}\| = 3\), \(\|\mathbf{b}\| = 4\), and \(\|\mathbf{a} + \mathbf{b}\| = 5\), find \(\|\mathbf{a} - \mathbf{b}\|\).
Problem 47. Find all vectors \(\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}\) such that \(\|\mathbf{x}\| = 1\) and \(\mathbf{x}\) is parallel to \(\mathbf{v} = \begin{bmatrix} 3 \\ -4 \end{bmatrix}\).
Problem 48. Given \(\mathbf{p} = \begin{bmatrix} 2 \\ -3 \\ 6 \end{bmatrix}\), find vectors \(\mathbf{q}\) and \(\mathbf{r}\) such that \(\mathbf{q}\) is a unit vector parallel to \(\mathbf{p}\) and \(\mathbf{r}\) is a unit vector antiparallel to \(\mathbf{p}\).
Problem 49. For vectors \(\mathbf{a} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}\), and \(\mathbf{c} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}\), verify that \(\|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 + \|\mathbf{c}\|^2 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})\).
Problem 50. If \(\mathbf{u}\) and \(\mathbf{v}\) are unit vectors such that \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{3}\), find \(\|\mathbf{u} - \mathbf{v}\|\).
Section F: Dot Products#
Problem 51. Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\) where \(\mathbf{a} = \begin{bmatrix} 2 \\ -3 \\ 4 \end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix} 1 \\ 5 \\ -2 \end{bmatrix}\).
Problem 52. Given \(\mathbf{u} = \begin{bmatrix} 3 \\ -1 \\ 2 \\ 4 \end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix} 2 \\ 6 \\ -1 \\ 0 \end{bmatrix}\), find \(\mathbf{u} \cdot \mathbf{v}\).
Problem 53. For vectors \(\mathbf{p} = \begin{bmatrix} 4 \\ 3 \end{bmatrix}\) and \(\mathbf{q} = \begin{bmatrix} -2 \\ 6 \end{bmatrix}\), calculate \((\mathbf{p} + \mathbf{q}) \cdot \mathbf{p}\).
Problem 54. If \(\mathbf{x} = \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}\) and \(\mathbf{y} = \begin{bmatrix} 4 \\ 1 \\ -1 \end{bmatrix}\), find the angle between \(\mathbf{x}\) and \(\mathbf{y}\) using the dot product formula.
Problem 55. Given unit vectors \(\hat{\mathbf{u}} = \begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}\) and \(\hat{\mathbf{v}} = \begin{bmatrix} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}\), find \(\hat{\mathbf{u}} \cdot \hat{\mathbf{v}}\) and determine if the vectors are orthogonal.
Section G: Linear Combinations#
Problem 56. Express the vector \(\mathbf{w} = \begin{bmatrix} 7 \\ -1 \\ 5 \end{bmatrix}\) as a linear combination of \(\mathbf{a} = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}\).
Problem 57. Determine if the vector \(\mathbf{v} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}\) can be written as a linear combination of \(\mathbf{u}_1 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}\) and \(\mathbf{u}_2 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\). If so, find the coefficients.
Problem 58. Given vectors \(\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}\), \(\mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}\), and \(\mathbf{e}_3 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\), find scalars \(c_1\), \(c_2\), and \(c_3\) such that \(c_1\mathbf{e}_1 + c_2\mathbf{e}_2 + c_3\mathbf{e}_3 = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}\).
Problem 59. Show that the vector \(\mathbf{z} = \begin{bmatrix} 7 \\ -2 \\ 12 \end{bmatrix}\) is a linear combination of \(\mathbf{p} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}\) and \(\mathbf{q} = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}\), and find all possible ways to express it.
Problem 60. (Challenging) Determine whether one of the vectors \(\mathbf{a} = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}\), and \(\mathbf{c} = \begin{bmatrix} 0 \\ 5 \\ -7 \end{bmatrix}\) can be expressed as a linear combination of the others.