Questions: The scalar product
Before attempting these questions, it is highly recommended that you read Guide: The scalar product, as well as Guide: Introduction to quadratic equations.
Q1
Find the scalar product of \(\mathbf{a}\) and \(\mathbf{b}\).
1.1. \(\mathbf{a}= \begin{bmatrix}6\\3\\4\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\4\\2\end{bmatrix}\)
1.2. \(\mathbf{a}= \begin{bmatrix}10\\-7\\4\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}3\\-5\\13\end{bmatrix}\)
1.3. \(\mathbf{a}= \begin{bmatrix}-44\\-12\\3\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}61\\-25\\93\end{bmatrix}\)
1.4. \(\mathbf{a}= \begin{bmatrix}54\\38\\0\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}32\\-55\\13\end{bmatrix}\)
1.5. \(\mathbf{a}= 2\mathbf{i} + 7\mathbf{j} + \mathbf{k}\) and \(\mathbf{b}= 6\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}\)
1.6. \(\mathbf{a}= -3\mathbf{i} + 10\mathbf{j} - 8\mathbf{k}\) and \(\mathbf{b}= \mathbf{i} - 12\mathbf{j} + 9\mathbf{k}\)
1.7. \(\mathbf{a}= 17\mathbf{j} + 23\mathbf{k}\) and \(\mathbf{b}= 6\mathbf{i} - 23\mathbf{j} - 8\mathbf{k}\)
1.8. \(\mathbf{a}= \mathbf{i}\) and \(\mathbf{b}= \mathbf{j}\).
What can you say about the result of 1.8.? Can you deduce similar conclusions for the scalar product of different combinations of the vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\)?
Q2
Using the geometric definition of the scalar products, find the smallest angle \(\theta\) in between \(\mathbf{a}\) and \(\mathbf{b}\) in degrees. If your answer is not a whole number, give your answer to an accuracy of one decimal place.
2.1. \(\mathbf{a}= \begin{bmatrix}-5\\2\\-3\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}2\\-2\\11\end{bmatrix}\)
2.2. \(\mathbf{a}= \begin{bmatrix}1\\1\\1\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\-1\\1\end{bmatrix}\)
2.3. \(\mathbf{a}= \begin{bmatrix}-8\\1\\-4\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}-1\\-5\\7\end{bmatrix}\)
2.4. \(\mathbf{a}= \begin{bmatrix}1.2\\-1.4\\-3.1\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}-5.4\\9.7\\-7.5\end{bmatrix}\)
2.5. \(\mathbf{a}= \begin{bmatrix}45\\65\\54\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}-19\\-58\\71\end{bmatrix}\)
2.6. \(\mathbf{a}= \begin{bmatrix}1\\0\\0\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}0\\0\\1\end{bmatrix}\)
2.7. \(\mathbf{a}= \begin{bmatrix}-1\\-2\\3\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}4\\-5\\6\end{bmatrix}\)
2.8. \(\mathbf{a}= \begin{bmatrix}-17\\3\\8\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}12\\-19\\-16\end{bmatrix}\)
Q3
Find the value(s) of \(\lambda\) for which \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular.
3.1. \(\mathbf{a}= \begin{bmatrix}2\\4\\7\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\\lambda\\-2\end{bmatrix}\)
3.2. \(\mathbf{a}= \begin{bmatrix}0\\1\\\lambda\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}1\\2\\3\end{bmatrix}\)
3.3. \(\mathbf{a}= \begin{bmatrix}9\\-2\\11\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}\lambda\\-\lambda\\3\end{bmatrix}\)
3.4. \(\mathbf{a}= \begin{bmatrix}\lambda\\6\\1\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}\lambda\\\lambda\\8\end{bmatrix}\)
3.5. \(\mathbf{a}= \begin{bmatrix}-2\lambda^2\\4\\14\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}3\\2\lambda\\1\end{bmatrix}\)
3.6. \(\mathbf{a}= \begin{bmatrix}-5\\9\\2\lambda\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}\lambda\\-2\\\lambda\end{bmatrix}\)
3.7. \(\mathbf{a}= \begin{bmatrix}-7\\4\\2\lambda\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}2\lambda\\1\\6\lambda\end{bmatrix}\)
3.8. \(\mathbf{a}= \begin{bmatrix}-25\\-\lambda^2\\-2\end{bmatrix}\) and \(\mathbf{b}= \begin{bmatrix}3\lambda\\-11\\7\end{bmatrix}\)
After attempting the questions above, please click this link to find the answers.
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.