The Cross Product Third Type of Multiplying Vectors

Cross Products

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

Finding Cross Products Using Equation

Evaluating a Determinant

Evaluating Determinants

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

Using Determinants to Find Cross Products

Algebraic Properties of the Cross Product If u, v, and w are vectors in space and if  is a scalar, then u x u = 0 u x v = -(v x u)  (u x v) = (  u) x v = u x (  v) u x (v + w) = (u x v) + (u x w)

Examples Given u = 2i – 3j + k v = -3i + 3j + 2k w = i + j + 3k Find (a) (3u) x v (b) v. (u x w)

Examples

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  where  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 a Vector Orthogonal to Two Given Vectors

Finding the Area of a Parallelogram Find the area of the parallelogram whose vertices are P 1 = (0, 0, 0), P 2 = (3,-2, 1), P 3 = (-1, 3, -1) and P 4 = (2, 1, 0) Two adjacent sides of this parallelogram are u = P 1 P 2 and v = P 1 P 3.

Finding the Area of the Parallelogram

Your Turn Try to do page 653 problems 1 – 47 odd.