Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Referring to the graph, what are the component forms of the following vectors?

Image Long Description

7. u
8. v
9. w
10. f
11. g
12. h

See Problem 2.

Transform each vector as described. Write the resulting vector in component form.

13. 〈5, 1〉; rotate 90 ° 90 degrees
14. 〈 − 2 , 3 〉 ; negative 2 comma 3 right pointing angle bracket semicolon rotate 180 ° 180 degrees
15. 〈0, 2〉; rotate 270 ° 270 degrees
16. 〈 11 , − 4 〉 ; left pointing angle bracket 11 comma negative 4 right pointing angle bracket semicolon  reflect across x-axis
17. 〈 − 3 , 0 〉 ; negative 3 comma 0 right pointing angle bracket semicolon reflect across y-axis
18. 〈4, 5〉; reflect across y = x y equals x
19. 〈 4 , − 3 〉 ; left pointing angle bracket 4 comma negative 3 right pointing angle bracket semicolon  reflect across y = − x y equals negative x

See Problems 3 and 4.

Let u = 〈 − 1 , negative 1 comma  3〉, v = 〈2, 4〉, and w = 〈2, − 5 negative 5 〉. Find the component forms of the following vectors.

20. u + v u plus v
21. v + w
22. u − v u minus v
23. u − w u minus w
24. 2u
25. − 4 w negative 4 w
26. 3 2 v 3 halves , v
27. − 3 v negative 3 v

See Problem 5.

Determine whether the vectors in each pair are normal to each other.

28. 〈6, − 3 negative 3 〉 and 〈2, 4〉
29. 〈8, − 4 negative 4 〉 and 〈 − 2 , 4 〉 negative 2 comma 4 right pointing angle bracket
30. [ 1 8 ] , matrix with 2 rows and 1 column , row1 column 1 , 1 , row2 column 1 , 8 , end matrix  and [ 4 − 2 ] . matrix with 2 rows and 1 column , row1 column 1 , 4 , row2 column 1 , negative 2 , end matrix
31. [ 0.8 − 0.6 ] . matrix with 2 rows and 1 column , row1 column 1 , 0.8 , row2 column 1 , negative 0.6 , end matrix  and [ 0.3 0.4 ] . matrix with 2 rows and 1 column , row1 column 1 , 0.3 , row2 column 1 , 0.4 , end matrix

Table of Contents