Vectors - Weebly



Vectors

Vectors are a useful concept in visually representing quantities of motion that would be difficult to represent otherwise. Because of their concise visual nature, we can use vectors to quickly represent and analyze a motion situation.

Vectors are quantities consisting of magnitude and direction. The examples of vectors that we'll be using in this topic are:

Quantity Example Magnitude Example Direction

displacement 13 meters 30º West of North

velocity 8m/sec 45º South of East

force 155 kg down

Vectors are represented by simply drawing an arrow:

The length represents the magnitude; therefore

it is necessary to choose a scale. We use

1 cm = __________ whatevers

Be careful to choose a scale that can "fit" the values to be represented into the amount of space available on your paper/page.

The direction of the vector is measured relative to north, south, east or west.

Vector Addition: We add vectors to find the result of 2 or more vectors of the same kind acting on an object. To do this, join he vectors head to tail without changing their length or direction. The resultant (R) is drawn so that its head touches a head. Some examples of vector addition are shown on page 2:

Example 1:

+ =

R

Example 2:

R

+ + =

Additional notes and examples:

Graphical Vector Problems

1. A plane flies east at 100km/hr. A wind blowing south at 30 km/hr

knocks the plane off course. Determine the resultant motion of the plane.

2. A boat sails east across a river at 8 km/h. The current is moving north (upstream) at 12 km/h and a breeze is blowing 25º N of E at 15 km/h. Find the boat's resultant motion.

3. John drives 30 km in a direction 57º N of W and then 21 km in a direction 20º E of N. What is his final displacement?

4. Three crazed shoppers are pulling a sweater (which is on sale) with the following forces: 10 N at 30º E of S

14 N at 18º W of N

17 N at 62º W of S

5. A plane flies 45º N of W at a speed of 1500 km/h. A wind is gusting at 180 km/h in a direction 30º N of E. Find the resultant motion of the plane.

6. A sailboat is headed downstream at 10m/s. A wind is blowing at 13m/s in a direction 56º E of S. Find the boat's resultant motion.

Vector Components: Once we have the resultant of a vector, we can further analyze it by determining its components. The components of a vector represent how much of the vector "acts" horizontally (vx) and how much of it acts vertically (vy). Visually, the components are the legs of a right triangle in which the resultant is the hypotenuse. The direction of the components is simply the horizontal and vertical directions already expressed in the resultant!!

Illustration of Vector components:

R R

vy vy

vx

vx

Problems: Determine the components of the resultants found in the Graphical Vector Problems on Pages 3 and 4.

Trigonometry: When a vector problem involves only a resultant and its components (which is often), the sketch of the problem will form a right triangle. If this is the case, we can use trig to solve the problem. This is convenient because rulers, protractors, and scales are NOT necessary!! The trig functions will generate the correct answers!

The Trig functions we use are:

sin ( = opposite/hypotenuse cos ( = adjacent/hypotenuse

tan ( = opposite/adjacent

* Note opposite and adjacent are with respect to angle (.

Trigonometry Problems

( R

b

(

a

1. If ( = 48º and a = 2, find b. 6. If ( = 53º and R = 5, find a.

2. If ( = 38º and b = 10, find R. 7. If a = 7 and R = 12, find (.

3. If a = 5 and b = 6, find (. 8. If b = 4 and R = 7, find (.

4. If ( = 37º and R = 6, find b. 9.If a = 9 and ( = 17º, find R.

5. If a = 5 and R = 15, find (. 10. If a = 3 and b = 7, find (.

Physics/Trig Problems

1. A boat sails at 20 km/hr in a direction 58º S of E. Find its components.

2. A plane tries to fly east but a 50 km/hr southerly wind knocks the plane off course by 17º. Find the plane's resultant speed and its original speed.

3. Find the components of a push applied to a stalled car if the push is 125 N in a direction 30º N of E.

4. A boat crossing a river is thrown off course by 20º by a 10 km/hr downstream current. Find the boat's original speed.

5. Suppose the displacement between your home and LHS is 4 km in a direction 48º N of W. What are the components of your displacement from home when you are at school?

6. A plane flying in a southwesterly direction has southern component of speed of 200 km/hr and a resultant speed of 350 km/hr. Find the plane's western component of speed and its exact direction.

7. A boat tries to cross a stream at 10 m/s. The current downstream is 3 m/s. Find the resultant speed and direction of the boat.

8. A plane tries to fly east at 137 km/hr but a north wind knocks the plane off course by 32º. Find the speed of the wind.

PARALLELOGRAM METHOD: To solve using the parallelogram method you can use the following equations - depending on what is given.

LAW of SINES

a = b = c

sin A sin B sin C

LAW of COSINES

a = ( b2 + c2 - 2bc cos A

b = (a2 + c2 - 2ac cos B

c = (a2 + b2 - 2ab cos C

Example:

A sailboat is headed downstream (south) at 10 m/s. A wind is blowing at 13 m/s in a direction 56( E of S. Find the boat's resultant motion.

a a = 10 b = 13 ................
................

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