7.2: Vectors in Three Dimensions
- Page ID
- 143584
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- Plot point \(P(2, 1, 3)\).
- Plot point \(Q(-3, 4, -2)\).
- Find the distance from point \(A(-5, -3, 5)\) to point \(B(0, 2, -2)\).
- Graph the surface \(x = 2\).
- Graph the surface \(y = -1\).
- Graph the surface \(z = 3\).
- Graph the surface \((x +1)(y - 3) = 0\).
- Graph the surface \(z(y^2 - 4) = 0\).
- Graph the surface \(xyz = 0\).
- Graph the surface \(y = x^2\).
- Graph the surface \(x = -z^2\)
- Graph the surface \(y^2 + z^2 = 1\).
- Graph the surface \( (x-1)^2 + (z + 2)^2 = 4 \).
- Graph the surface \( \dfrac{y^2}{9} + \dfrac{z^2}{4} = 1 \).
- Graph the surface \( \dfrac{z^2}{16} - \dfrac{x^2}{4} = 1\).
- Graph the surface \( x^2 + y^2 + z^2 = 1\).
- Graph the surface \( (x - 1)^2 + (y + 2)^2 + (z + 1)^2 = 4\).
- Find the center and radius of the sphere \(x^2 + y^2 + z^2 + 2x + 2z + 1 = 0\).
- Consider the points \(P(4, -1, 1)\) and \(Q(3, 1, -1)\). Express the vector \(\vec{PQ}\) in component form.
- Consider the points \(A(-1, 0, 1)\) and \(B(3, -2, -4)\). Express the vector \(\vec{BA}\) in terms of the standard unit vectors.
- Sketch vector \(\mathbf{v} = (1, 2, -3)\) in standard position, then sketch vector \(\mathbf{v}\) again with a different initial point.
- Sketch vector \(\mathbf{u} = -2\mathbf{i} - \mathbf{j} + \mathbf{k}\) in standard position, then sketch vector \(\mathbf{u}\) again with a different initial point.
- Perform the vector arithmetic. Express your answer in component form.
\begin{equation*}
(2, 8, 7) + (-5, 6, -6)
\end{equation*}
- Perform the vector arithmetic. Express your answer in terms of the standard unit vectors.
\begin{equation*}
(4\mathbf{i} - 6\mathbf{j} - \mathbf{k}) + (7\mathbf{i} - 10\mathbf{k})
\end{equation*}
- Perform the vector arithmetic. Express your answer in component form.
\begin{equation*}
(-10, 0, 10) - (3, -2, -5)
\end{equation*}
- Perform the vector arithmetic. Express your answer in terms of the standard unit vectors.
\begin{equation*}
(6\mathbf{i} + 2\mathbf{j} - 9\mathbf{k}) - (9\mathbf{i} + 2\mathbf{j} - 10\mathbf{k})
\end{equation*}
- Let vector \(\mathbf{u} = (4, -4, 2)\), vector \(\mathbf{v} = (-2, -3, 1)\), and vector \(\mathbf{w} = (-5, -5, 4)\). Calculate \(\mathbf{r} = -2\mathbf{w} - 3(\mathbf{u} - 3\mathbf{v})\). Express your answer in component form.
- Let vector \(\mathbf{u} = -2\mathbf{j} - \mathbf{k}\), vector \(\mathbf{v} = 4\mathbf{i} - 5\mathbf{j} - 5\mathbf{k}\), and vector \(\mathbf{w} = -4\mathbf{i} + \mathbf{j} + 4\mathbf{k}\). Calculate \(\mathbf{r} = -3\mathbf{v} - 2(\mathbf{u} - 2\mathbf{w})\). Express your answer in terms of the standard unit vectors.
- Find the norm of vector \(\mathbf{u} = (-1, 3, -5)\).
- Find the magnitude of vector \(\mathbf{w} = 4\mathbf{i} - 5\mathbf{j} - \mathbf{k}\).
- Find a unit vector \(\mathbf{u}\) pointing in the same direction as vector \(\mathbf{v} = (1, -4, 5)\). Express \(\mathbf{u}\) in component form.
- Normalize the vector \(\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\). Express your answer in terms of the standard unit vectors.
- Let vector \(\mathbf{w} = (-7, 1, 0)\). Calculate \(-5\mathbf{w}\). Express your answer in component form.
- Find a vector \(\mathbf{u}\) in the same direction as \(\mathbf{v} = 3\mathbf{i} - 4\mathbf{j} - 5\mathbf{k}\) with magnitude \(\|\mathbf{u}\| = 2\). Express your answer in terms of the standard unit vectors.
- Find the equation of the sphere that has a diameter with endpoints \((-2, -1, 5)\) and \((-6, 5, 3)\).