Questions: Vector addition and scalar multiplication
Before attempting these questions, it is highly recommended that you read Guide: Vector addition and scalar multiplication.
Q1
Answer the following questions.
1.1. If \(\mathbf{a} = 4\mathbf{i} + 5\mathbf{j} + 7\mathbf{k}\) and \(\mathbf{b} = 8\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\), find \(\mathbf{a} + \mathbf{b}\).
1.2. If \(\mathbf{a} = 3\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{b} = 2\mathbf{i} + 5\mathbf{k}\), find \(\mathbf{a} + \mathbf{b}\).
1.3. If \(\mathbf{a} = -2\mathbf{i} + 6\mathbf{k}\) and \(\mathbf{b} = -4\mathbf{i} + 11\mathbf{j} -8\mathbf{k}\), find \(\mathbf{a} - \mathbf{b}\).
1.4. If \(\mathbf{a} = 4\mathbf{i} + 12\mathbf{j} -7\mathbf{k}\), \(\mathbf{b} = 3\mathbf{i} -3\mathbf{j} -2\mathbf{k}\) and \(\mathbf{c} = 11\mathbf{i} -4\mathbf{j} +9\mathbf{k}\), find \(\mathbf{a} - (\mathbf{b} + \mathbf{c})\).
Q2
Solve the following, expressing your answers in terms of the unknown scalars \(x,y,z\).
2.1. If \(\mathbf{a} = \begin{bmatrix}x \\2y\\0\end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix}3x \\5y\\0\end{bmatrix}\), find \(\mathbf{a} + \mathbf{b}\).
2.2. If \(\mathbf{a} = \begin{bmatrix}5 \\3y\\5z\end{bmatrix}\) and \(\mathbf{b} = \begin{bmatrix}-2\\2x\\6z\end{bmatrix}\), find \(\mathbf{a} - \mathbf{b}\).
2.3. If \(\mathbf{a} = \begin{bmatrix}2x \\3y\\4z\end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix}-2x\\y\\0\end{bmatrix}\) and \(\mathbf{c} = \begin{bmatrix}0\\4y\\4z\end{bmatrix}\), find \(\mathbf{a} + \mathbf{b} - \mathbf{c}\).
2.4. If \(\mathbf{a} = \begin{bmatrix}2x\\3y\\5z\end{bmatrix}\), what is \(\mathbf{a} + \mathbf{0}\)?
Q3
Answer the following questions.
3.1. If \(\mathbf{u} = 5\mathbf{j} + 6\mathbf{k}\), find \(3\mathbf{u}\).
3.2. If \(\mathbf{v} = \begin{bmatrix}0\\-3\\7\end{bmatrix}\), find \(-6\mathbf{v}\).
3.3. If \(\mathbf{u} = \begin{bmatrix}0\\5\\6\end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix}0\\-3\\7\end{bmatrix}\), find \(4\mathbf{v} - 3\mathbf{u}\).
3.4. If \(\mathbf{u} = \begin{bmatrix}0\\5\\6\end{bmatrix}\), \(\mathbf{v} = \begin{bmatrix}0\\-3\\7\end{bmatrix}\) and \(\mathbf{w} = \begin{bmatrix}2\\3\\-4\end{bmatrix}\), find \(-2\mathbf{w} - (4\mathbf{u} -2\mathbf{v})\).
Q4
Answer the following questions.
4.1. If \(A = (3,4,5)\). \(B = (-2,5,7)\), find \(\overrightarrow{AB}\).
4.2. If \(A = (2,5,7)\), \(B = (6,11,7)\) and \(C = (0,1,2)\), find \(\overrightarrow{AB} - \overrightarrow{AC}\).
4.3. If \(\overrightarrow{AB} = \begin{bmatrix}6\\7\\-2\end{bmatrix}\) and \(B = (1,5,9)\), find the coordinates of \(A\).
4.4. If \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{b} = 3\mathbf{i} -5\mathbf{j}\), find \(13\mathbf{i} -9\mathbf{j}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
4.5. If \(\mathbf{a} = \begin{bmatrix}3\\5\\z\end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix}-1\\-3\\4\end{bmatrix}\) and \(2\mathbf{a} + 3\mathbf{b} = \begin{bmatrix}x\\y\\0\end{bmatrix}\), solve for the unknown scalars \(x,y,z\).
4.6. Given that \(\mathbf{a}\) and \(\mathbf{b}\) are parallel, if \(\mathbf{a} = (x-7)\mathbf{i} + (5x+1)\mathbf{k}\) and \(\mathbf{b} = -2\mathbf{i} + 8\mathbf{k}\), find \(x\).
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 Renee Knapp, Kin Wang Pang as part of a University of St Andrews STEP project.
- v1.1: edited 05/24 by tdhc.