Show that the equations x2 y2 z2 z ge 0 represents an

Question

3D Geometry and Vectors - Math 53, section 213

1. What 3D figure does the equation x² + z² ≤ 9 represent? Sketch it on coordinate axes.

2. Show that the equations x² + y² = z², z ≥ 0 represents an infinite, hollow cone pointing upwards along the z-axis.

3. Write down equations that describe an ice cream cone: that is, a filled in cone with a solid hemisphere sitting on top.

4. Find the cosine of the angle between the vectors (1, 2, 3) and (4, 5, 6) in the coordinate plane.

5. Find the angle ∠ABC where A = (1, 0, 0), B = (0, 1, 0), and C = (0, 0, 1).

6. Compute the cross product of (1, 1, -1) and (2, 4, 6). What is the area of the parallelogram spanned by these vectors?

7. Use the scalar product a · (b × c) to derive the formula for the volume of a rectangular prism (box) and verify that it is indeed the product of the length, width, and height.

8. What is the volume of the parallelepiped spanned by (1, 2, 3), (4, 5, 6), and (1, 3, 6)?