1. Vector Refresher Part 2 Vector Addition Vector Addition Multiplication by Scalar Quantities Multiplication by Scalar Quantities Graphical Representations Graphical Representations Analytical Methods Analytical Methods

2. Vector Addition A lot of times, it’s important to know the resultant of a set of vectorsA lot of times, it’s important to know the resultant of a set of vectors The resultant vector can be described as the sum of all the vector quantities you wish to evaluateThe resultant vector can be described as the sum of all the vector quantities you wish to evaluate We’ll look at graphical and analytical techniques for doing thisWe’ll look at graphical and analytical techniques for doing this

3. Graphical Methods For Vector Addition 2 ways to look at vector addition graphically2 ways to look at vector addition graphically Parallelogram methodParallelogram method Connect the tails of both vectors to start.

4. Graphical Methods For Vector Addition 2 ways to look at vector addition graphically2 ways to look at vector addition graphically Parallelogram methodParallelogram method Create a parallelogram with each vector as one of the sides.

5. Graphical Methods For Vector Addition 2 ways to look at vector addition graphically2 ways to look at vector addition graphically Parallelogram methodParallelogram method The resultant goes from the place where 2 tails meet to the place where 2 heads meet

6. Graphical Methods For Vector Addition 2 ways to look at vector addition graphically2 ways to look at vector addition graphically Head-to-tail methodHead-to-tail method With this method, the vectors are added by placing the tail of a vector at the head of another one to make a continuous path.

7. Graphical Methods For Vector Addition 2 ways to look at vector addition graphically2 ways to look at vector addition graphically Head-to-tail methodHead-to-tail method The addition of these vectors results in a vector that starts at the tail of the first vector and ends at the head of the final vector

8. Graphical Methods For Vector Addition 2 ways to look at vector addition graphically2 ways to look at vector addition graphically Head-to-tail methodHead-to-tail method If we had a 3 rd vector, we could add it to the previous result

9. Graphical Methods For Vector Addition 2 ways to look at vector addition graphically2 ways to look at vector addition graphically Head-to-tail methodHead-to-tail method If we had a 3 rd vector, we could add it to the previous result

10. Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed.

11. Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed. Multiplying a vector by 2 yields a vector with the SAME DIRECTION and twice the length

12. Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed. Multiplying a vector by 1/2 yields a vector with the SAME DIRECTION and half the length

13. Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed. Multiplying a vector by -1 yields a vector with the OPPOSITE DIRECTION and the same length

14. Example Draw Start by multiplying vector A by 1/2

15. Example Draw

16. Example Draw Next, multiply vector B by -3

17. Example Draw Now add these vectors together

18. Example Draw The resultant vector is shown in green

19. Analytical Method for Vector Addition When vectors are added to each other, the components in each direction are summed.

20. Analytical Method for Vector Addition When vectors are added to each other, the components in each direction are summed.

21. Analytical Method for Vector Addition When vectors are added to each other, the components in each direction are summed.

22. Multiplying a Vector By a Scalar When multiplying a vector by a scalar, the scalar must be applied to each component of the vector Thus, if we multiply by a scalar “k”, we get:

23. The first thing to do is determine F 3 Example Determine the resultant force caused by the following 3 force vectors.

24. The first thing to do is determine F 3 Example Determine the resultant force caused by the following 3 force vectors.

25. The first thing to do is determine F 3 Example Determine the resultant force caused by the following 3 force vectors.

26. The first thing to do is determine F 3 Example Determine the resultant force caused by the following 3 force vectors.

27. The first thing to do is determine F 3 Example Determine the resultant force caused by the following 3 force vectors.

28. The first thing to do is determine F 3 Example Determine the resultant force caused by the following 3 force vectors.

29. The first thing to do is determine F 3 Example Determine the resultant force caused by the following 3 force vectors.

30. Now we can add the 3 vectors together to determine the resultant force Example Determine the resultant force caused by the following 3 force vectors.

31. Now we can add the 3 vectors together to determine the resultant force Example Determine the resultant force caused by the following 3 force vectors.

32. Now we can add the 3 vectors together to determine the resultant force Example Determine the resultant force caused by the following 3 force vectors.

33. Don’t forget the units, they’re as important to the answer as the numbers are Example Determine the resultant force caused by the following 3 force vectors.

34. Don’t forget the units, they’re as important to the answer as the numbers are Example Determine the resultant force caused by the following 3 force vectors.