Overview
1. Given the points (3, 8, 1), (1, 1, 1), and (x, 5, 6), where x is a real number:
- Compute a value of x such that the points form a right triangle.
2. Consider the vectors u = ⟨3, 4, 8⟩ and v = ⟨1, −4, −2⟩:
- Determine if u × v is orthogonal to both u and v by computing u · (u × v).
3. The vectors u = ⟨−3, a, 4⟩ and v = ⟨4, 5, 1⟩ are orthogonal:
- Determine the value of a.
- Compute a third vector that is orthogonal to both u and v.
- Is your vector the only vector that is orthogonal to both u and v? Explain why or why not.
4. Find a set of parametric equations of a line with the following characteristics:
- The line passes through the point (−3, 5, 7).
- The line is parallel to both the xz-plane and the yz-plane.
5. Find a set of parametric equations of a line with the following characteristics:
- The line passes through the point (5, 2, −6).
- The line is parallel to the vector v = 3i − 4j.
6. Find the equation of a plane that passes through the points (3, 4, 5), (−2, −9, 0), and (4, 4, 3).
7. Find two vectors that are normal to the plane: 3x − 9y + z − 27 = 0
8. Verify that the point (3, 1, 0) is on the plane 3x − 9y + z − 27 = 0.
- If the point is on the plane, find a vector that passes through (3, 1, 0) and is orthogonal to the plane.
9. Find the domain of the vector-valued function: r(t) = (1 / (t − 1))i + √(t² − 1)j + ln(t)k
10. Compute the derivative r′(t) of the vector-valued function from Problem 9.
11. Compute the domain of r′(t) from Problem 10.