- Section 12.4
- u × v = |u||v| sin(a) n
- Non-zero u, v parallel iff u×v = 0
- Determinant formulas
- You can also remember how to calculate cross product as a criss-cross
pattern and a few rules
- Torque vector = |r||F| sin(a) n
- |(u×v)⋅w| = |u×v| |w| cos(a)
- Properties in book say...
- Scalar coefficients can factor out of cross products
- Cross product distributes over addition
- Zero (the vector) crossed with anything is the zero vector
- But cross product is not commutative, u×v and v×u point in opposite directions
- Cross product isn’t associative either
- What’s going on in the triangle area example in book?
- Magnitude of cross product is area of parallelogram, and triangle
is half a parallelogram
- Magnitude of cross product is area of parallelogram, and triangle
is half a parallelogram
- Examples
- ⟨ 1, 2, 3 ⟩ × ⟨ 1, -1, 2 ⟩
- = ⟨ (2×2 - 3×(-1)), (3×1-1×2), (1×(-1)-2×1) ⟩
- = ⟨ 7, 1, -3 ⟩
- Writing vectors above each other can help visualize criss-cross pattern
- ⟨1,2,3⟩
- ⟨1,-1,2⟩
- Find “right” vector for yesterday’s video game scenario
- Gaze vector n = ⟨ 1/2, 0, √3/2 ⟩
- Up vector v = ⟨ -√3/4, 0, 1/4 ⟩
- Right vector, u = n × v = ⟨ 0, -(1/8 - -3/8), 0 ⟩ = ⟨ 0, -1/2, 0 ⟩
- ⟨ 1, 2, 3 ⟩ × ⟨ 1, -1, 2 ⟩
- Problem set
- Lines
- Read section 12.5 through “The Distance from a Point to a
Line in Space”
- Top of page 720 through first 1/4 page 723