Inner Product and Orthogonality

  1. Given \( \overrightarrow{w}=\left[\begin{array}{r}3 \\ -1 \\ -5\end{array}\right], \: \overrightarrow{x}=\left[\begin{array}{r}6 \\ -2 \\ 3\end{array}\right]\). Compute: (a) \(\overrightarrow{w} \cdot \overrightarrow{w}, \quad \overrightarrow{x} \cdot \overrightarrow{w}, \: \)  and  \(\:\dfrac{\overrightarrow{x} \cdot \overrightarrow{w}}{\overrightarrow{w} \cdot \overrightarrow{w}}\quad \) (b)  \(\left(\dfrac{\overrightarrow{x} \cdot \overrightarrow{w}}{\overrightarrow{x} \cdot \overrightarrow{x}}\right) \overrightarrow{x} \quad \) (c) \(\|\overrightarrow{x}\|\)  solution

    2.  

  2. Find a unit vector in the direction of the given vector. \(\quad \overrightarrow{u}=\left[\begin{array}{r} -6 \\ 4 \\ -3 \end{array}\right] \).   solution

    3.  

  3. Show that \(\left\{\overrightarrow{u}_{1}, \overrightarrow{u}_{2}\right\}\) is an orthogonal basis for \(\mathbb{R}^{2}\). Then express \(\overrightarrow{x}\) as a linear combination of the \(\overrightarrow{u}\)'s. \( \quad \overrightarrow{u}_{1}=\left[\begin{array}{r}3 \\ 1\end{array}\right], \overrightarrow{u}_{2}=\left[\begin{array}{r}-2 \\ 6\end{array}\right],\) and \(\overrightarrow{x}=\left[\begin{array}{r} -6\\ 3\end{array}\right].\)   solution

    4.  

  4. Compute the orthogonal projection of \(\left[\begin{array}{l}1 \\ 7\end{array}\right]\) onto the line through \(\left[\begin{array}{r}-4 \\ 2\end{array}\right]\) and the origin.   solution

    5.  

  5. Let \(W\) be a subspace of \(\mathbb{R}^{n},\) and let \(W^{\perp}\) be the set of all vectors orthogonal to \(W\). Verify the three conditions to show that \(W^{\perp}\) is a subspace of \(\mathbb{R}^{n}\).  solution

    6.