For Exercises 7-14, for the given vectors, find b. v• v c. w•w (See Example 1) a. v.w 7. v = (-3, 5), w = (10, –8) 8. v = (6, –7), w = 3(-1, -4) 3 2 1 1 4 5 9. v = 10. v = W3= --- W3= -- 8' 3 6' 5 11. v = 6i, w = -3j 12. v = 2j, w = 14i 13. v = = 2.li – 6.8j, w = 0.4i – 0.3j 14. v = -0.5i + 1.6j, w = -0.li + 2.3j

Take this test to review the material in Chapters 4and Chapters 5. After you are finished, check your work against the answers in the back of the book. Write w=(7,2,4) as a linear combination of the vectors v1, v2 and v3 if possible. v1=(2,1,0), v2=(1,1,0), v3=(0,0,6)

2. If , and the vector is drawn with its tail at the point, find the coordinates of the point at the head of .

We mentioned in Section 7.5 that our algebraic treatment of vectors could be attributed, in part, to the Irish mathematician William Rowan Hamilton. Hamilton considered his greatest achievement to be the discovery of qualernions, which he (incorrectly) predicted would revolutionize physics. Research the subject of quaternions. What are they? Why did quaternions fail to be as useful for physics as Hamilton predicted? How are quaternions useful, instead, for 3.D computer graphics? Write a paragraph or two about your findings.