Problems

Vector Operations and Dot Products

Problem 1

Consider the following vectors:

  1. Find the coordinates of \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\)
  2. Find \(\mathbf{u} + \mathbf{v} + \mathbf{w}\) (does the answer make geometric sense?)
  3. Find \(\mathbf{u} - \mathbf{v} - \mathbf{w}\) (does the answer make geometric sense?)
  4. Normalize the vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\). That is, find a vector in the same direction with norm equal to one.
  5. Check that \(\mathbf{u}\) and \(\mathbf{v}\) satisfy Cauchy-Schwartz inequality.

Problem 2

Let \(\mathbf{v}\), \(\mathbf{w}\), \(\mathbf{z} \in \mathbb{R}^n\). Suppose that \(\mathbf{v} \cdot \mathbf{w} = 2\), \(\mathbf{v} \cdot \mathbf{z} = -1\), \(\mathbf{z} \cdot \mathbf{w} = 1\), \(\|\mathbf{v}\| = \sqrt{3}\), \(\|\mathbf{w}\| = 2\), and \(\|\mathbf{z}\| = \sqrt{5}\). Find the following:

  1. \((2\mathbf{v} + 3\mathbf{w}) \cdot \mathbf{z}\)
  2. \(\mathbf{v} \cdot (\mathbf{v} - 2\mathbf{w} + 3\mathbf{z})\)
  3. \((3\mathbf{v} - 4\mathbf{z}) \cdot (\mathbf{w} + 5\mathbf{z})\)
  4. \(\|\mathbf{v} + \mathbf{w}\|\)
  5. \(\|\mathbf{v} - \mathbf{w}\|\)
  6. \(\|2\mathbf{v} - 6\mathbf{z}\|\)
  7. Find the angle between \(\mathbf{v}\) and \(\mathbf{z}\)
  8. Find \(c \in \mathbb{R}\) such that \(\mathbf{v}\) is orthogonal to \(\mathbf{w} + c\mathbf{z}\)
  9. Find \(c \in \mathbb{R}\) such that \(\mathbf{w} - c\mathbf{v}\) is orthogonal to \(\mathbf{v}\)

Problem 3

Let \(\mathbf{v}\), \(\mathbf{w} \in \mathbb{R}^n\). Suppose that \(\|\mathbf{v}\| = 2\), \(\|\mathbf{v} + \mathbf{w}\| = \sqrt{3}\) and \(\|\mathbf{w}\| = \sqrt{2}\). Find \(\mathbf{v} \cdot \mathbf{w}\).

Problem 4

Let \(\mathbf{v}\), \(\mathbf{w} \in \mathbb{R}^n\). Suppose that \(\|\mathbf{v}\| = \sqrt{2}\), \(\|\mathbf{v} - \mathbf{w}\| = \sqrt{3}\) and \(\mathbf{v} \cdot \mathbf{w} = \frac{1}{2}\). Find \(\|\mathbf{w}\|\).

Problem 5

Suppose that \(\mathbf{u} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}\)

  1. Find a constant \(c \in \mathbb{R}\) such that \(\mathbf{v} - c\mathbf{u}\) is orthogonal to \(\mathbf{u}\). Name this vector \(\mathbf{f_2} = \mathbf{v} - c\mathbf{u}\).
  2. Find constants \(c, d \in \mathbb{R}\) such that \(\mathbf{w} - c\mathbf{u} - d\mathbf{f_2}\) is orthogonal to \(\mathbf{u}\) and orthogonal to \(\mathbf{f_2}\). Name this vector \(\mathbf{f_3} = \mathbf{w} - c\mathbf{u} - d\mathbf{f_2}\).
  3. Find constants \(c, d, e \in \mathbb{R}\) such that the vectors \(c\mathbf{u}\), \(d\mathbf{f_2}\), and \(e\mathbf{f_3}\) have norm one. Rename these vectors \(\mathbf{e_1} = c\mathbf{u_1}\), \(\mathbf{e_2} = d\mathbf{f_2}\), and \(\mathbf{e_3} = e\mathbf{f_3}\) and write them explicitly.

Problem 6

Let \(\mathbf{v_1}\), \(\mathbf{v_2}\), \(\mathbf{v_3}\), \(\mathbf{v_4}\) be vectors satisfying:

\(\mathbf{v_i} \cdot \mathbf{v_j} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{otherwise} \end{cases}\)

  1. Find \((2\mathbf{v_1} - 3\mathbf{v_2}) \cdot (2\mathbf{v_3} + 4\mathbf{v_4})\)
  2. Find \((\mathbf{v_1} + \mathbf{v_2}) \cdot (\mathbf{v_1} - \mathbf{v_2})\)
  3. Find \(\|\mathbf{v_4}\|\)
  4. Find \(\|4\mathbf{v_1} - 3\mathbf{v_2}\|\)
  5. Find \(\|2\mathbf{v_1} - 3\mathbf{v_2} + 4\mathbf{v_3} - 5\mathbf{v_4}\|\)