1 / 20

11.3 The Dot Product

11.3 The Dot Product. Geometric interpretation of dot product. A dot product is the magnitude of one vector times the portion of the vector that points in the same direction as that vector (the projection in the direction of the other.). u = (4,10) v = (-2,3) u = (1,5,7) v = (-1,3,4).

sierra-richard
Download Presentation

11.3 The Dot Product

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. 11.3 The Dot Product

  2. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points in the same direction as that vector (the projection in the direction of the other.)

  3. u = (4,10) v = (-2,3) u = (1,5,7) v = (-1,3,4) Note: To find a dot product on the TI89 Press 2nd 5 (math) – 4 matrix – L Vector ops – 3 dotP dotP([1,5,7],[-1,3,4]) Find the dot product of the given vectors

  4. Note property 5 is explained on the next slide.

  5. __ We can see that using the Pythagorean Theorem yields the same result as √a∙a So we can write the length or magnitude of a vector in terms of the dot product. This will be important in the second semester.

  6. Angle Between Two Vectors Proven on last slide

  7. Alternative form of dot product

  8. Notes: This definition will allow us to expand the notion of orthogonal to higher Dimensions. (This will be important next semester in Linear Algebra.) Orthogonal and perpendicular are generally used interchangeably. However there is a subtle difference. Perpendicular means that two items (planes, lines segments vectors … whatever) must meet to make a 90 degree angle… However, orthogonal includes this situation plus includes the zero vector is orthogonal to all other vectors even though we could not say that it is perpendicular to all other vectors.

  9. Find the angle between u and v • u = (3,-1,3) • v =(-4,0,2)

  10. What is meant by the angle between two vectors?

  11. Determine if the given vectors are orthogonal • u = (3,-1,2) • w =(1,-1,3)

  12. Determine if the given vectors are orthogonal u = (3,-1,2) w =(1,-1,3) u and v are not orthogonal because the dot product is not 0. What value x will make vectors u and q orthogonal? q = (1,-1,x)

  13. Example 5 Find the projection of u onto v and the vector component of u orthogonal to v u = 3i – 5j +2kv = 7i + j -2k

  14. Note solve this problem 3 ways: Solve with special right triangles Solve with Trigonometry and force times distance Solve with the dot product

  15. "A mathematician is a device for turning coffee into theorems“ -- P. Erdos

  16. Proof of the dot product

More Related