1. Adding, Subtracting and Parallel Vectors Lessons 10.5 – 10.6

2. Opposite of a vector  - v (the opposite of v) is the vector with the same length but opposite direction.  If v is a vector with polar representation [r, θ] then – v = [r, θ + 180]  Sum of two vectors: + = Difference of two vectors: - =

3. Example 1  A person is exerting a force of 80 pounds in a direction 22˚ east of north in order to move a sofa. Another person was exerting a force of 60 pounds due north. Describe the vector representing the total force x = 80 cos 68 + 60 cos 90 y = 80 sin 68 + 60 sin 90 length: Direction: Total force of about 137.5 pounds in a direction 12.6˚ east of north.

4. Example 2  If a person did not want the sofa moved and was trying to counter the force described in example 1, what magnitude force would the person need to exert and in what direction? It is currently being moved with a force of 137.5 pounds in a direction 12.6˚ east of north. Opposite theorem: 137.5 pounds in a direction 12.6˚ west of south.

5. Parallel Vectors and Equations of Lines  A vector w = is a scalar multiple of vector u = written w = ku iff there exists a real number k such that =  Nonzero vectors u and v are parallel iff there exists a real number t ≠ 0 st u = tv  The line through P = that is parallel to the vector v = has the vector equation = t  The line through P = that is parallel to the vector v = has parametric equations x = x 0 + ta y = y 0 + tb Where t may be any real number.

6. Example 1  The vector u begins a (3, -7) and ends at (-4, 8). The vector v = 6.6u is in standard position. Where does v begin and where does it end?  It is in standard position, so we know it begins at (0, 0)  Vector u =  V = 6.6  V =

7. Example 2  Tell whether the two given vectors are parallel and justify your answer.  and Yes, k = -6  [13, 17˚] and [5, 163˚] No, they are not opposite directions.

8. Example 3 Give an example of a vector parallel to that also meets the following criteria.  Has opposite direction:  Has magnitude equal to 1:

9. Example 2  In one population model, the population of a city is expected to be 300,000 in 2025 and is expected to decline by 2000 people a year each year after that.  A. Find parametric equations for the line, where t is the number of years after 2025, x is the year and p is the population in that year. x(t) = 2025 + t y(t) = 300,000 – 2000t b. Find a vector equation for this line.