1. Addition and Subtraction of Vectors Lecture V1.1 Example 1 Moodle

2. Vectors have a magnitude and direction

3. The resultant vector R = A + B Shift BShift A R = A + B = B + A commutative law Addition of Vectors

4. A + B + C D = A + B + C D = (A + B) + C D = A + (B + C) D = A + B + C = (A + B) + C = A + (B + C) associative law

5. R = A + B R + (-A) = B R - A = B Subtraction of Vectors

6. D = A - B

7. Unit Vectors and Coordinate Systems A x = A x i A y = A y j A = A x i + A y j. A x = A cos  A y = A sin 

8. Addition of Vectors by Components R = R x i + R y j = (A x + B x ) i + (A y + B y ) j R x = A x + B x R y = A y + B y

9. 3-Dimensional Vectors A = A x i + A y j + A z k

10. 3-Dimensional Vectors A x = R cos  A y = R sin  A z = A cos  R = A sin  A x = A sin  cos  A y = A sin  sin  A z = A cos 

11. Direction Cosines = 1

12. Given the vectors A = i – 2j + 4k and B = 3i + j - 2k, find R = A + B. Matlab Example 1 >> A = [1 -2 4] A = 1 -2 4 >> B = [3 1 -2] B = 3 1 -2 >> R = A + B R = 4 -1 2 >>

13. Use column vectors >> A = [1; -2; 4] A = 1 -2 4 >> B = [3; 1; -2] B = 3 1 -2 >> R = A + B R = 4 2 >>

14. Find A = |A| and B = |B| for the vectors A = i – 2j + 4k and B = 3i + j - 2k Matlab Example 2 >> A = [1 -2 4] A = 1 -2 4 >> magA = norm(A) magA = 4.5826 >> B = [3 1 -2] B = 3 1 -2 >> magB = norm(B) magB = 3.7417 >>

15. Find the unit vector A in the direction of the vector A = i – 2j + 4k Matlab Example 3 >> A = [1 -2 4] A = 1 -2 4 >> lambdaA = A/norm(A) lambdaA = 0.2182 -0.4364 0.8729 >>

16. Find the direction cosines of the vector A = i – 2j + 4k Matlab Example 4 >> A = [1 -2 4] A = 1 -2 4 >> l = A(1)/norm(A) l = 0.2182 >> m = A(2)/norm(A) m = -0.4364 >> n = A(3)/norm(A) n = 0.8729 >>