The Cross Product Third Type of Multiplying Vectors
Determinants • It is much easier to do this using determinants because we do not have to memorize a formula. • Determinants were used last year when doing matrices • Remember that you multiply each number across and subtract their products
Using Determinants to Find Cross Products • This concept can help us find cross products. • Ignore the numbers included in the column under the vector that will be inserted when setting up the determinant.
Using Determinants to Find Cross Products • Find v x w given • v = i + j • w = 2i + j + k
Using Determinants to Find Cross Products • If v = 2i + 3j + 5k and w = i + 2j + 3k, • find • (a) v x w • (b) w x v • (c) v x v
Algebraic Properties of the Cross Product • If u, v, and w are vectors in space and if a is a scalar, then • u x u = 0 • u x v = -(v x u) • a(u x v) = (au) x v = u x (av) • u x (v + w) = (u x v) + (u x w)
Examples • Given u = 2i – 3j + kv = -3i + 3j + 2k • w = i + j + 3k • Find • (a) (3u) x v • (b) v . (u x w)
Geometric Properties of the Cross Product • Let u and v be vectors in space • u x v is orthogonal to both u and v. • ||u x v|| = ||u|| ||v|| sin q, where q is the angle between u and v. • ||u x v|| is the area of the parallelogram having u ≠ 0 and v ≠ 0 as adjacent sides
Geometric Properties of the Cross Product • u x v = 0 if and only if u and v are parallel.
Finding a Vector Orthogonal to Two Given Vectors • Find a vector that is orthogonal to • u = 2i – 3j + k and v = i + j + 3k • According to the preceding slide, u x v is orthogonal to both u and v. So to find the vector just do u x v
Finding a Vector Orthogonal to Two Given Vectors • To check to see if the answer is correct, do a dot product with one of the given vectors. Remember, if the dot product = 0 the vectors are orthogonal
Finding the Area of a Parallelogram • Find the area of the parallelogram whose vertices are P1 = (0, 0, 0), • P2 = (3,-2, 1), P3 = (-1, 3, -1) and • P4 = (2, 1, 0) • Two adjacent sides of this parallelogram are u = P1P2 and v = P1P3.
Your Turn • Try to do page 653 problems 1 – 47 odd.