Question

1. Find the midpoint of the segment with ends (4, -3, 1) and (-2, 5, 3).

2. Determine whether the vectors u=3i+j-2k and v=2i-4j+k are orthogonal.

3. Describe and sketch 9x^2 + 9y^2 - 4z^2=0

Answer 1

1.

$$x_M=\dfrac{4-2}{2}=1,$$

$$y_M=\dfrac{-3+5}{2}=1,$$

$$z_M=\dfrac{1+3}{2}=2$$

$$M(1,1,2)$$

2.

$$\vec u\cdot\vec v=3(2)+1(-4)+(-2)(1)=0$$

Then the vectors $\vec u=3\vec i+\vec j-2\vec k$ and $\vec v=2\vec i-4\vec j+\vec k$ are orthogonal.

3.

$$9x^2 + 9y^2 - 4z^2=0$$

Cone

$$\dfrac{z^2}{9}=\dfrac{x^2}{4}+\dfrac{y^2}{4}$$

Horizontal traces are circles.

Vertical traces in the planes $x=k$ and $y=k$ are hyperbolas if $k\not=0$ but are pairs of lines if $k=0.$