Then I saw “a new heaven and a new earth,” for the first heaven and the first earth had passed away, and there was no longer any sea. I saw the Holy City, the new Jerusalem, coming down out of heaven from God, prepared as a bride beautifully dressed for her husband. And I heard a loud voice from the throne saying, “Look! God’s dwelling place is now among the people, and he will dwell with them. They will be his people, and God himself will be with them and be their God. Revelation 21:1-3 NIV
11-Review
Take this test as you would take a test in class. When you are finished, check your work against the answers. On this assignment round your answers to three decimal places unless otherwise directed.
- Plot the points A(1, –2, 3), B(0, 3, –5), and C(–2, 0, 2)
Use the points A(1, –2, 3), B(0, 3, –5), and C(–2, 0, 2). Let \(\overset{\rightharpoonup}{u}\) be the vector from A to B and \(\overset{\rightharpoonup}{v}\) be the vector from B to C.
- Is the triangle formed isosceles?
- Find the midpoint of segment AC.
- Find the standard form of the equation of a sphere with B as the center and C on the surface of the sphere.
- Find the parametric equations for the line passing through A and B.
- Find the general form of the equation of the plane passing through the points A, B, and C.
- Find the distance between point A and the plane x + 2y − z − 4 = 0.
- Write \(\overset{\rightharpoonup}{u}\) and \(\overset{\rightharpoonup}{v}\) in component form.
- Find \(\| \overset{\rightharpoonup}{u} \|\).
- Find a unit vector in the direction of \(\overset{\rightharpoonup}{u}\).
- Find \(\overset{\rightharpoonup}{u} \cdot \overset{\rightharpoonup}{v}\).
- Find \(\overset{\rightharpoonup}{u} × \overset{\rightharpoonup}{v}\).
- Find \(\overset{\rightharpoonup}{u} × \overset{\rightharpoonup}{u}\).
- Find the angle between \(\overset{\rightharpoonup}{u}\) and \(\overset{\rightharpoonup}{v}\).
- Use vectors to find the area of the parallelogram with vertices (–2, –2), (4, –1), (0, 5), and (6, 6).
- Determine whether \(\overset{\rightharpoonup}{n} = 2\hat{i} + \hat{j} − 4\hat{k}\) and \(\overset{\rightharpoonup}{m} = \hat{i} + 2\hat{j} + \hat{k}\) are parallel, orthogonal, or neither.
- Determine whether \(\overset{\rightharpoonup}{n} = 5\hat{i} + 2\hat{j} − 2\hat{k}\) and \(\overset{\rightharpoonup}{m} = -3\hat{i} + \hat{j} + 2\hat{k}\) are parallel, orthogonal, or neither.
- Find the volume of the parallelepiped.
- Plot the intercepts and graph the plane 4x + 4y + 3z − 12 = 0
- Plot the intercepts and graph the plane 6x + 3y − 4z + 12 = 0
Answers
- No, it is scalene.
- \(\left(-\frac{1}{2}, -1, \frac{5}{2}\right)\)
- x² + (y − 3)² + (z + 5)² = 62
- \(\left\{\begin{align} x &= 1 - t \\ y &= -2 + 5t \\ z &= 3 - 8t\end{align}\right.\)
- 11x + 23y + 13z − 4 = 0
- \(\frac{5\sqrt{6}}{3}\)
- \(\overset{\rightharpoonup}{u} = \langle -1, 5, -8 \rangle, \overset{\rightharpoonup}{v} = \langle -2, -3, 7 \rangle\)
- \(3\sqrt{10}\)
- \(\langle -\frac{\sqrt{10}}{30}, \frac{\sqrt{10}}{6}, -\frac{4\sqrt{10}}{15} \rangle\)
- –69
- \(\langle 11, 23, 13 \rangle\)
- \(\langle 0, 0, 0 \rangle\)
- 157.47°
- 40
- orthogonal
- neither
- 512
