1 / 12

Vectors in Space

Vectors in Space. We live in a Three Dimensional World. Rectangular Coordinates in Space. Right handed coordinate system We now have an ordered triple (x,y,z) associated with each point. Graphing examples. Representing Vectors in Space.

alijah
Download Presentation

Vectors in Space

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Vectors in Space We live in a Three Dimensional World

  2. Rectangular Coordinates in Space • Right handed coordinate system • We now have an ordered triple (x,y,z) associated with each point. • Graphing examples

  3. Representing Vectors in Space • Since we have a new axes, we will now need a third unit vector to represent the z axis. • i = (1, 0, 0) j = (0, 1, 0) and • k = (0, 0, 1)

  4. Position Vector • To find the position vector, we will now have • v = (a2 – a1)i + (b2 – b1)j + (c2 – c1)k

  5. Addition, Subtraction and Scalar Multiplication • All rules that applied in two dimensions, now apply in three dimensions

  6. Unit Vector in Direction of v • For any non zero vector v, the vector • is a unit vector that has the same direction as v.

  7. Dot Product • We find the dot product the same way we found it in two dimensions, we just add the third dimension

  8. Angle Between Two Vectors • We use the same formula we used in two dimensions including the third dimension

  9. Direction Angles of Vectors in Space • This is the only truly new operation. • There are three direction angles • a = angle between v and the positive x-axis, 0 ≤ a ≤ p • b = angle between v and the positive y-axis, 0 ≤ b ≤ p • g = angle between v and the positive z-axis, 0 ≤ g ≤ p

  10. Direction Angles

  11. Direction Cosines • The direction cosines play the same role in space as slope does in the plane.

  12. Property of Direction Cosines • If a, b, and g are the direction angles of a nonzero vector v in space, then • cos2a + cos2b + cos2g = 1

More Related