1. Objectives: Graph vectors in 3 dimensions Use cross products and dot products to find perpendicular vectors

2. Vectors in 3D Space Ordered triples(x, y, z) z y x Locate the point (3, 5, 4)

3. Vectors Draw the vector from the origin to the point (3, -2, 6). Draw the vector from the origin to the point (-2, 5, 4) Find the ordered triple that represents the vector from A(3, 7, -1) to B(10, -4, 0). (x 2 – x 1, y 2 – y 1, z 2 – z 1 )

4. Magnitude The force of the wind blowing the John Hancock Building at one moment can be expressed as the vector (3120 N, 195 N, 5 N). What is the magnitude of the force? |x|=

5. Add, Subtract, Multiply by a Scalar Find an ordered triple to represent u in each equation if v = (1, -3, -8) and w = (3, 9, -1). u = v + w u = 4v + 3w u = 5v – 2w

6. Unit Vectors Rewrite a vector as the sum of unit vectors. (3, -5, 8) = i = (1, 0, 0) j = (0, 1, 0) k = (0, 0, 1) First, express GH as an ordered triple. Then write as the sum of unit vectors for G(10, -3, 15) and H(4, 1, -11).

7. Practice 1. Locate (3, 4, 9). Then find the magnitude of the vector from the origin. 2. Locate (-2, 1, 3). Then find the magnitude of the vector from the origin. 3. For A(-2, 5, 8) and B(3, 9, -3), find an ordered triple that represents AB. Then find the magnitude of the vector. 4. Write (9, 3, -2) as the sum of unit vectors. 5. If v = (1, -3, -8) and w = (3, 0, 1), find 4v – 3w.

8. Perpendicular Vectors Two vectors are perpendicular if and only if their inner product is zero. 2Da b = x 1 x 2 + y 1 y 2 3D a b = x 1 x 2 + y 1 y 2 + z 1 z 2 Which of the following vectors are perpendicular? a = (3, 12)b = (8, -2) c = (3, -2) v = (-6, 2, 10) w = (4, 1, 3)

9. To find a perpendicular vector... Cross Product a x b = Given a = (5, 2, 3) and b = (-2, 5, 0), find the cross product. Then verify that the vector is perpendicular. a x b =

10. Practice 1. Find the inner product and state whether the vectors are perpendicular. (-6, 1) (-1, 2) (3, -2, 4) ( 1, -4, 0) 2.Find the cross product. Then verify that the resulting vector is perpendicular to the given vectors. (1, 3, 2) x (2, -1, -1) Assignment: 3D Worksheet