Three Dimensional Geometry Questions and Answers

Answer: C

The Gaussian curvature is always = 0.

The elliptic cylinder is generated by straight lines parallel to the z-axis through the ellipse \(x^{2}/2 + y^{2}/3 = 1\). Along each ruling the normal curvature is 0, so one principal curvature is 0 everywhere, and \(K = \kappa_{1}\kappa_{2} = 0\).

Equivalently, the cylinder can be unrolled onto a plane without stretching, so it is locally isometric to the plane and by Gauss's Theorema Egregium has the plane's curvature, zero. The other options describe curved surfaces such as ellipsoids (positive) or saddles (negative).

Answer: A

The direction cosines are \(\frac{2}{3}, -\frac{1}{3}, \frac{-2}{3}\).

Divide each direction ratio by the length of the vector (2, -1, -2):

\[ \sqrt{2^{2} + (-1)^{2} + (-2)^{2}} = \sqrt{9} = 3 \]

giving 2/3, -1/3, -2/3, whose squares add to 1. Dividing by 9 (the square of the length) gives option B, whose squares do not add to 1. Option C drops the minus signs, and option D changes the sign of only some terms, so it does not point along the same line.

Answer: B

k = 2.

The direction ratios are the denominators: (2k, 3, -1) for the first line and (8, 6, -2) for the second. (The first line's '2+2' is a misprint for z+2; it does not affect the denominators.) Parallel lines have proportional direction ratios:

\[ \frac{2k}{8} = \frac{3}{6} = \frac{-1}{-2} = \frac{1}{2} \]

so 2k = 4 and k = 2. k = -2 reverses the sign of the first ratio, and 1/2 or 4 make 2k/8 equal to 1/8 or 1, out of proportion with the other two.

Answer: B

The angle is \(\pi/2\).

The direction ratios are (7, -5, 1) and (1, 2, 3). The cosine of the angle is proportional to their dot product:

\[ 7(1) + (-5)(2) + 1(3) = 7 - 10 + 3 = 0 \]

A zero dot product means the lines are perpendicular, so the angle is π/2. An angle of 0 would need proportional direction ratios, and π/3 or π/4 would need a non-zero dot product.

Answer: C

The point lies in the 7th octant.

The usual numbering of the octants by the signs of (x, y, z) is: I (+, +, +), II (-, +, +), III (-, -, +), IV (+, -, +), V (+, +, -), VI (-, +, -), VII (-, -, -) and VIII (+, -, -).

All three coordinates of (-1, -5, -7) are negative, so it lies in octant VII. The 1st octant has all coordinates positive, the 2nd has only x negative, and the 8th has y and z negative but x positive.