Analytic Geometry Answers

A cable hangs in a parabolic arc between two poles 100 feet apart. The poles are 30 feet high and the lowest point on the suspended cable is 5 feet above the ground. Find the equation of the arc if the vertex is the lowest point of the cable.

Also, find the height of the cable at a point 10 feet from one of the poles.

If C = [2.5 cm, 80°], i.e., the magnitude and direction of C are 2.5 cm and 80°, D = [3.5 cm,

120°], and E = D – 2C, then what is the direction of E (to the nearest degree)?

: P is a point on the parabola whose ordinate

equals its abscissa. A normal is drawn to the parabola at

P to meet it again at Q. If S is the focus of the parabola

then the product of the slopes of SP and SQ is

Write down the equation of the plane containing the point (2 , 3 , 4 ) and parallel to

the vectors

u= (2 , -3 ,2) and v= (0 , 1 , 2) .

1.1 Find the value of 𝑐 for which the vectors (𝑐, 1, 1) and (−1, 2, 0) are orthogonal 

(perpendicular).

Consider two vectors a and b. Select all the correct statements above the two vectors above. 

1. a is perpendicular to a×b.
2. a×b is parallel to b×a.
3. If a×b=0 then a=0 or b=0.
4. If a and b are parallel vectors then "\\mathbf{a}\\cdot\\mathbf{b}=|\\mathbf{a}||\\mathbf{b}|"
5. If a and b are parallel vectors then a×b=0.

You are given the two vectors "\\mathbf{a}=(1,2,1)" and "\\mathbf{b}=(3,-4,2)" . Suppose that the pair of unit vectors perpendicular to both a and b are given by

"\\pm \\frac{1}{\\sqrt{p}}(q,1,r),"

where p, q and r are some constants.

Determine the values of p, q and r.

Given the equation of a sphere "x^{2}+y^{2}+z^{2}-4x+2y+2z=-5"

Which of the following is the vector equation of the sphere above?

1. "|\\mathbf{r}-(2,1,1)|=1"
2. "|\\mathbf{r}-(2,-1,-1)|=1"
3. "|\\mathbf{r}-(2,-1,1)|=1"
4. "\u2223r\u2212(2,\u22121,1)\u2223=1."
5. "\u2223r\u2212(2,1,\u22121)\u2223=1."

Consider the vectors "\\mathbf{a}=(-k, 1+k, k(1+k))" and "\\mathbf{b}=(1+k, k(1+k), -k)" where k is some constant. Which of the following is/are correct?

1. a and b are perpendicular vectors for any k
2. ∣a∣=∣b∣.
3. a and b are unit vectors for k=1
4. The values of k for which ∣a∣=1 are 0 and -1
5. The two vectors above are parallel.