Answers: The scalar product
These are the answers to Questions: The scalar product.
Please attempt the questions before reading these answers!
Q1
1.1. For \(\mathbf{a}= \begin{bmatrix}6\\3\\4\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\4\\2\end{bmatrix}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = 26\).
1.2. For \(\mathbf{a}= \begin{bmatrix}10\\-7\\4\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}3\\-5\\13\end{bmatrix}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = 117\).
1.3. For \(\mathbf{a}= \begin{bmatrix}-44\\-12\\3\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}61\\-25\\93\end{bmatrix}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = -2237\).
1.4. For \(\mathbf{a}= \begin{bmatrix}54\\38\\0\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}32\\-55\\13\end{bmatrix}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = -362\).
1.5. For \(\mathbf{a}= 2\mathbf{i} + 7\mathbf{j} + \mathbf{k}\) and \(\mathbf{b}= 6\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = 48\).
1.6. For \(\mathbf{a}= -3\mathbf{i} + 10\mathbf{j} - 8\mathbf{k}\) and \(\mathbf{b}= \mathbf{i} - 12\mathbf{j} + 9\mathbf{k}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = -195\).
1.7. For \(\mathbf{a}= 17\mathbf{j} + 23\mathbf{k}\) and \(\mathbf{b}= 6\mathbf{i} - 23\mathbf{j} - 8\mathbf{k}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = -575\).
1.8. For \(\mathbf{a}= \mathbf{i}\) and \(\mathbf{b}= \mathbf{j}\), the scalar product is \(\mathbf{a}\cdot\mathbf{b} = 0\).
As the scalar product of \(\mathbf{a}= \mathbf{i}\) and \(\mathbf{b}= \mathbf{j}\) is \(0\), they are perpendicular to each other. This is true for any combination of any distinct pair of \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\). However, since any vector is parallel to itself, it follows that \(\mathbf{i}\cdot \mathbf{i} = |\mathbf{i}||\mathbf{i}| = |1||1| = 1\); similar results hold for \(\mathbf{j}\cdot \mathbf{j}\) and \(\mathbf{k}\cdot \mathbf{k}\).
Q2
2.1. For \(\mathbf{a}= \begin{bmatrix}-5\\2\\-3\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}2\\-2\\11\end{bmatrix}\), the angle \(\theta\) is \(132.2^\circ\).
2.2. For \(\mathbf{a}= \begin{bmatrix}1\\1\\1\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\-1\\1\end{bmatrix}\), the angle \(\theta\) is \(70.5^\circ\).
2.3. For \(\mathbf{a}= \begin{bmatrix}-8\\1\\-4\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}-1\\-5\\7\end{bmatrix}\), the angle \(\theta\) is \(108.7^\circ\).
2.4. For \(\mathbf{a}= \begin{bmatrix}1.2\\-1.4\\-3.1\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}-5.4\\9.7\\-7.5\end{bmatrix}\), the angle \(\theta\) is \(86.2^\circ\).
2.5. For \(\mathbf{a}= \begin{bmatrix}45\\65\\54\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}-19\\-58\\71\end{bmatrix}\), the angle \(\theta\) is \(95.1^\circ\).
2.6. For \(\mathbf{a}= \begin{bmatrix}1\\0\\0\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}0\\0\\1\end{bmatrix}\), the angle \(\theta\) is \(90^\circ\).
2.7. For \(\mathbf{a}= \begin{bmatrix}-1\\-2\\3\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}4\\-5\\6\end{bmatrix}\), the angle \(\theta\) is \(43.0^\circ\).
2.8. For \(\mathbf{a}= \begin{bmatrix}-17\\3\\8\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}12\\-19\\-16\end{bmatrix}\), the angle \(\theta\) is \(137.8^\circ\).
Q3
3.1. For \(\mathbf{a}= \begin{bmatrix}2\\4\\7\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\\lambda\\-2\end{bmatrix}\) to be perpendicular, then \(\lambda= 3\).
3.2. For \(\mathbf{a}= \begin{bmatrix}0\\1\\\lambda\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\2\\3\end{bmatrix}\) to be perpendicular, then \(\displaystyle\lambda= -\frac{2}{3}\).
3.3. For \(\mathbf{a}= \begin{bmatrix}9\\-2\\11\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}\lambda\\-\lambda\\3\end{bmatrix}\) to be perpendicular, then \(\lambda= -3\).
3.4. For \(\mathbf{a}= \begin{bmatrix}\lambda\\6\\1\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}\lambda\\\lambda\\8\end{bmatrix}\) to be perpendicular, then \(\lambda= -2\) or \(\lambda= -4\).
3.5. For \(\mathbf{a}= \begin{bmatrix}-2\lambda^2\\4\\14\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}3\\2\lambda\\1\end{bmatrix}\) to be perpendicular, then \(\displaystyle\lambda = \frac{7}{3}\) or \(\lambda= -1\).
3.6. For \(\mathbf{a}= \begin{bmatrix}-5\\9\\2\lambda\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}\lambda\\-2\\\lambda\end{bmatrix}\) to be perpendicular, then \(\displaystyle\lambda= \frac{9}{2}\) or \(\lambda= -2\).
3.7. For \(\mathbf{a}= \begin{bmatrix}-7\\4\\2\lambda\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}2\lambda\\1\\6\lambda\end{bmatrix}\) to be perpendicular, then \(\displaystyle\lambda= \frac{2}{3}\) or \(\displaystyle\lambda= \frac{1}{2}\).
3.8. For \(\mathbf{a}= \begin{bmatrix}-25\\-1\lambda^2\\-2\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}3\lambda\\-11\\7\end{bmatrix}\) to be perpendicular, then \(\lambda= 7\) or \(\displaystyle\lambda= -\frac{2}{11}\).
Version history and licensing
v1.0: initial version created 08/23 by Ritwik Anand as part of a University of St Andrews STEP project.
- v1.1: edited 05/24 by tdhc.