3D Vectors Day 3

1. 3D Vectors Day 3

2. Ex. 1 Find a unit vector that is orthogonal to both: We want this to be a unit vector, so find the magnitude of a X b: A unit vector in the direction of a X b is:

3. Another often used formula is for finding the magnitude of a cross product: where is the angle between a and b One application of the cross product is that it can be used in finding the area of a parallelogram! Area of parallelogram with a and b as adjacent sides

4. Ex. 2 Show that the quadrilateral with vertices at the following points is a parallelogram and find its area: A (5,2,0) B (2,6,1) C (2,4,7) D (5,0,6) is a parallelogram Since AB is adjacent to BC, lets find the magnitude of its cross product to get the area:

5. In geometry, the area of a parallelogram is given by: So the area of a triangle is given by: With vectors since the area of a parallelogram is given by: (where a and b are adjacent sides) So this can similarly be extended to the area of a triangle: a b

6. Ex. 3 Find the area of given A (-1,2,3), B (2,1,4), and C (0,5,-1)

7. Triple Scalar Product So what can we use this for?

8. bXc Volume of a Parallelepiped a b c Absolute value- just to make it positive! Let’s prove this: The base is a parallelogram. Base area = The height of the parallelogram is given by: Volume = (height) (base area)

9. Ex. 4 Find the volume of a parallelepiped having as adjacent edges. = 36

10. Test for Coplanar Points If you are given four points in space, there are two possibilities: 1) Either those four points are in different planes and form a tetrahedron 2) Or those four points are coplanar. Four points A, B, C, and D that have position vectors a, b, c, d are coplanar iff: Notice this is really just the triple scalar product!

11. Ex. 5 Are the points A(1,2,-4), B(3,2,0), C(2,5,1) and D(5,-3,-1) coplanar? are coplanar!