Answers to the Exercises - 3D Math Primer for Graphics and Game Development - page 758

B.3 Chapter 3

(Page 109.)

1. (a) Object space.
(b) We could compare my world-space x-coordinate with the topic's world-space x-coordinate.
Or, we just examine the sign of the upright-space x-coordinate.
(c) World space.
(d) Object space. Or you might say that we could take a dot product with our facing
direction vector—which is equivalent to extracting the object-space z-coordinate.
2. First translate the point by [−12, 0, 6] relative to the axes, and then rotate clockwise around
the y-axis 42 o .

3. (a) Linearly dependent. The middle basis vector is the zero vector, which cannot belong

to a linearly independent set because it can be expressed as a product of any other
basis vector and 0.
(b) Linearly independent.
(c) Linearly dependent. For 3D vectors, the largest linearly independent set we could
hope for is three vectors, but this set has four.
(d) Linearly dependent. The last vector is a multiple of the first.
(e) Linearly dependent. The last vector is the sum of the first two.
(f) Linearly independent.

4. (a) Orthogonal.

(b) Not orthogonal. All of the pairs of vectors have nonzero dot products.
(c) Orthogonal.
(d) Orthogonal.
(e) Not orthogonal. The first pair of vectors is perpendicular, but [7,−1, 5][−2, 0, 1] =
−9, and [5, 5,−6][−2, 0, 1] = −16.

5. (a) No. The second and third basis vectors clearly do not have unit length.

(b) No. None of the basis vectors have unit length.
(c) Yes, they are orthonormal.
(d) No. The first and second basis vectors are not perpendicular.
(e) Yes, they are orthonormal.
(f) Yes, they are orthonormal.
(g) No. The second and third basis vectors do not have unit length.

6. (a) Upright: (−0.866, 2.000, 0.500); World: (0.134, 12.000, 3.500)

(b) Upright: (0.866, 2.000,−0.500); World: (1.866, 12.000, 2.500)
(c) Upright: (0, 0, 0); World: (1, 10, 3)
(d) Upright: (1.116, 5.000,−0.067); World: (2.116, 15.000, 2.933)