Section 9.4: The Cross Product Practice HW from Stewart Textbook (not to hand in) p. 664 # 1, 7-17.

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Presentation transcript:

Section 9.4: The Cross Product Practice HW from Stewart Textbook (not to hand in) p. 664 # 1, 7-17

Cross Product of Two Vectors The cross product of two vectors produces a vector (unlike the dot product which produces s scalar) that has important properties. Before defining the cross product, we first give a method for computing a determinant.

Definition: The determinant of a matrix, denoted by, is defined to be the scalar

Example 1: Compute Solution:

Definition: If a = i + j + k and b = i + j + k be vectors in 3D space. The cross product is the vector i + j + k

To calculate the cross product more easily without having to remember the formula, we using the following “determinant” form.

We calculate the determinant as follows: (note the alternation in sign)

Example 2: Given the vectors and. Find a.

b.

c.

Properties of the Cross Product Let a, b, and c be vectors, k be a scalar. 1. Note! k ( k k b) 4. 5.

Geometric Properties of the Cross Product Let a and b be vectors 1. is orthogonal to both a and b.

Example 3: Given the vectors and, show that the cross product is orthogonal to both a and b.

Note:,, x y z

2. 3. if and only a = k b, that is, if the vectors a and b are parallel. 4 gives the area having the vectors a and b as its adjacent sides.

Example 4: Given the points P(0, -2, 0), Q(-1, 3, 4), and R(3,0,6). a.Find a vector orthogonal to the plane through these points. b.Find the area of the parallelogram with the vectors and as its adjacent sides. c. Find area of the triangle PQR. Solution: (In the typewritten notes)