7.1: Vectors in the Plane
- Page ID
- 143371
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- Sketch vector \(\mathbf{v} = (-1, 3)\) in standard position, then sketch vector \(\mathbf{v}\) again with a different initial point.
- Sketch vector \(\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}\) in standard position, then sketch vector \(\mathbf{a}\) again with a different initial point.
- Sketch vector \(\mathbf{u} = \left(\dfrac{9}{2}, -4\right)\) in standard position, then sketch vector \(\mathbf{u}\) again with a different initial point.
- Sketch vector \(\mathbf{w} = -\dfrac{3}{2}\mathbf{j}\) in standard position, then sketch vector \(\mathbf{w}\) again with a different initial point.
- Consider the points \(A(2, 0)\) and \(B(-2, 1)\). Express the vector \(\vec{AB}\) in component form. Plot the points and the vector.
- Consider the points \(P(3, 2)\) and \(Q(-4, -3)\). Express the vector \(\vec{QP}\) in terms of the standard unit vectors. Plot the points and the vector.
- Perform the vector arithmetic. Express your answer in component form.
\begin{equation*}
(-4, 2) + (-9, -9)
\end{equation*}
- Perform the vector arithmetic. Express you answer in terms of the standard unit vectors.
\begin{equation*}
(\mathbf{i} - 2\mathbf{j}) + (6\mathbf{i} + \mathbf{j})
\end{equation*}
- Vectors \(\mathbf{u}\) and \(\mathbf{v}\) are plotted tip-to-tail below. Using the plot, sketch \(\mathbf{w} = \mathbf{u}+\mathbf{v}\).
- Perform the vector arithmetic. Express your answer in component form.
\begin{equation*}
(-3, 3) - (-8, 2)
\end{equation*}
- Perform the vector arithmetic. Express you answer in terms of the standard unit vectors.
\begin{equation*}
(-6\mathbf{i} + 8\mathbf{j}) - (-9\mathbf{i})
\end{equation*}
- Vectors \(\mathbf{u}\) and \(\mathbf{v}\) are plotted below with the same initial point. Using the plot, sketch vector \(\mathbf{w} = \mathbf{u}-\mathbf{v}\).
- Let vector \(\mathbf{u} = (3, -2)\). Calculate \(-3\mathbf{u}\) and express your answer in component form. Plot \(\mathbf{u}\) and \(-3\mathbf{u}\) on the same coordinate plane.
- Let vector \(\mathbf{b} = 5\mathbf{i} - 4\mathbf{j}\). Calculate \(\dfrac{1}{2}\mathbf{b}\) and express your answer in terms of the standard unit vectors. Plot \(\mathbf{b}\) and \(\dfrac{1}{2}\mathbf{b}\) on the same coordinate plane.
- Let vector \(\mathbf{u} = (-2, 2)\), vector \(\mathbf{v} = (-4, 5)\), and vector \(\mathbf{w} = (-2, -5)\). Calculate \(\mathbf{r} = -2\mathbf{w} - 5(\mathbf{v} - 3\mathbf{u})\). Express your answer in component form.
- Let vector \(\mathbf{u} = 4\mathbf{i} + 2\mathbf{j}\), vector \(\mathbf{v} = -\mathbf{i} + \mathbf{j}\), and vector \(\mathbf{w} = -5\mathbf{i} - \mathbf{j}\). Calculate \(\mathbf{r} = 3(\mathbf{v} - 5\mathbf{u}) - 4\mathbf{w}\). Express your answer in terms of the standard unit vectors.
- Find the norm of vector \(\mathbf{v} = (-2, 4)\).
- Find the magnitude of vector \(\mathbf{w} = 6\mathbf{i} - 3\mathbf{j}\).
- Find a unit vector \(\mathbf{u}\) pointing in the same direction as vector \(\mathbf{v} = (-10, -2)\). Express \(\mathbf{u}\) in component form.
- Normalize vector \(\mathbf{v} = -6\mathbf{i} + 8\mathbf{j}\). Express your answer in terms of the standard unit vectors.
- Find a vector \(\mathbf{a}\) in the same direction as vector \(\mathbf{b} = (8, 9)\) with magnitude \(\|\mathbf{a}\| = 2\). Express your answer in component form.
- Find a vector \(\mathbf{u}\) in the same direction as vector \(\mathbf{v} = \mathbf{i} - 7\mathbf{j}\) with magnitude \(\|\mathbf{u}\| = 3\). Express your answer in terms of the standard unit vectors.
- Find vector \(\mathbf{w}\) so that \(\|\mathbf{w}\| = 3\) and the angle between \(\mathbf{w}\) and the positive \(x\)-axis is \(\theta = \dfrac{5\pi}{4}\). Express your answer in component form.
- Find vector \(\mathbf{v}\) so that \(\|\mathbf{v}\| = 10\) and the angle between \(\mathbf{v}\) and the positive \(x\)-axis is \(\theta = \dfrac{2\pi}{3}\). Express your answer in
terms of the standard unit vectors.
- Find the angle \(\theta \in [0, 2\pi)\) that vector \(\mathbf{u} = (-4\sqrt{2}, 4\sqrt{2})\) makes with the positive \(x\)-axis.
- Find the angle \(\theta \in [0, 2\pi)\) that vector \(\mathbf{a} = -2\sqrt{3}\mathbf{i} - 2\mathbf{j}\) makes with the positive \(x\)-axis.
