# Vectors

### The process of determining the magnitude of a vector is known as vector resolution. The two methods of vector resolution are:

• parallelogram method

• trigonometric method

### The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components (sides of the parallelogram) using the scale. The parallelogram method of vector resolution performed as follows:

• select a scale and accurately draw the vector to scale in the indicated direction.

• sketch a parallelogram around the vector: beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a parallelogram.

• draw the components of the vector; the components are the sides of the parallelogram; be sure to place arrowheads on these components to indicate their direction (up, down, left, right).

• label the components of the vectors with symbols to indicate which component is being represented by which side; a northward velocity component might be labeled vN ; etc.

• measure the length of the sides of the parallelogram and use the scale to determine the magnitude of the components in real units; label the magnitude on the diagram.

### The trigonometric method of vector resolution involves using trigonometric functions to determine components of the vector. It can be used to determine the length of the sides of a right triangle if one angle and the length of one side are known. This method is as follows:

• construct a sketch of the vector in the indicated direction; label its magnitude and the angle which it makes with the horizontal.

•  draw a rectangle about the vector such that the vector is the diagonal of the rectangle; beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a parallelogram.

• draw the components of the vector; the components are the sides of the rectangle; be sure to place arrowheads on these components to indicate their direction.

• label the components of the vectors with symbols to indicate which component is being represented by which side; a northward force component would be labeled F-north and so forth.

• use the sine function to determine the length of the side opposite the indicated angle; substitute the magnitude of the vector for the length of the hypotenuse; use some algebra to solve the equation for the length of the side opposite the indicated angle.

• repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

### Review Questions:

1. Hess takes flying lessons and is flying 80 mph. What is his speed if he encounters a 10 mph headwind?

70mph

2. What is Hess's speed if he encounters a 10 mph tailwind?

90 mph

3. What is Hess's speed if he encounter s a 10 mph crosswind?

80.6 mph

4. Ranada travels in a boat at 5 m/s East and encounters a current traveling 2.5 m/s, North. What is the resultant velocity of the motor boat?

5.59 m/s

5. If the width of the river is 80 meters wide, how much time does it take for Ranada to travel from shore to shore?

16.0 seconds

6. If Mathew is running with the football at 5 m/s East for a touchdown and a tackle from the opposing team pushes him 1.2 m/s, North. He is 80 meters from the endzone and 20 meters from the north sideline. Will he score a touchdown?

7. Suppose Jocelyn goes hang gliding and tries to land on the bulls eye of a target 120 meters straight ahead of her. She glides at 6 m/s, east (straight ahead) and encounters an air current of 3.8 m/s, south. How far off will she be, if any, from her target? At what angle would she need to adjust her next glide to hit the target in the center assuming all factors remain the same?

8. Trevor decides to go out for the swim team. To qualify, he must swim across the Colorado at a point that is 120 meters wide with an average speed of at least 7 m/s. The coach is unaware of currents and vectors. He swims at 6 m/s with a cross current of 3.8 m/s. Will Trevor qualify? Were there any factors that affected his qualifying or not qualifying?

20 seconds Yes. The cross current gave him a speed of 7.1 m/s which was faster than he was able to swim enabling him to qualify!

9. Suppose the current were 5 m/s in problem #8, how much time would be required for Trevor to swim across the same river? How far would he travel downstream during this time?

It would still take the same time but he would end up 100 m downstream!

10. Find the magnitude and direction for the resultant of the two forces: F1 = 100 N at an angle of 40 north of west and F2 = 200 N at an angle of30 west of south.

R = 208 N; angle 211.6 degrees

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