Adding Vectors by the Component Method

Feel free to use to accompanying notes sheet.

Adding Vectors by the Component Method

Yesterday we added vectors which were at right angles to one another. What would happen if the vectors were not at right angles?

A crow (who apparently isn’t aware that he should only fly in straight lines) first flies 10 km at 60° N of E. The he flies 25 km at 10° N of E. What is his total displacement?

Adding Vectors by the Component Method – The Strategy

A = 10 km @ 60° N of E

B = 25 km @ 10° N of E

Obviously the answer is not 35 km, so what must be done?

60°

10°

Adding Vectors by the Component Method

A = 10 km @ 60° N of E

B = 25 km @ 10° N of E

Obviously the answer is not 35 km, so what must be done?

Find the RESULTANT

R = resultant

Adding Vectors by the Component Method

The vectors need to be added vectorally.

Both vectors need to be RESOLVED into their components.

The components are then added together to find the resultant.

A to be resolved into Ax and Ay.

B to be resolved into Bx and By.

Adding Vectors by the Component Method – Resolve each vector into components

A = 10 km @ 60° N of E

B = 25 km @ 10° N of E

Adding Vectors by the Component Method – Resolve each vector into components

A = 10 km @ 60° N of E

B = 25 km @ 10° N of E

Ay

Ax

By

Bx

Adding Vectors by the Component Method

Ax = 10cos60° Ay = 10sin60°

Bx = 25cos10° By = 25sin10°

A = 10 km @ 60° N of E

B = 25 km @ 10° N of E

Ay

Ax

By

Bx

Adding Vectors by the Component Method

The x-components and y-components can each be considered legs of the resulting triangle.

Ay

Ax

ByBx

Adding Vectors by the Component Method

The x-components and y-components can each be considered legs of the resulting triangle.

Ay

Ax

By

Bx

Adding Vectors by the Component Method – Construct the “resulting triangle” from components

Rx = Ax + Bx Ry = Ay + By

Ay

Ax

By

Bx

R

Rx

Ry

Adding Vectors by the Component Method

How do we find R?

YES!! Pythagorean Theorem: R2 = Rx2 + Ry

2

Ay

Ax

By

Bx

R

Rx

Ry

Rx = 29.60

Ry = 13.00

R = ??

Adding Vectors by the Component Method

R2 = RX2 + RY

2

Ay

Ax

By

Bx

R

Rx

Ry

Adding Vectors by the Component Method

Math – IN DEGREE MODE Ax = 10cos60° = 5.00 km

Ay = 10sin60° = 8.66 km

Bx = 25cos10° = 24.6 km

By = 25sin10° = 4.34 km

Rx = Ax + Bx = 29.6 km

Ry = Ay + By = 13.00 km R = 32.33 km

Adding Vectors by the Component Method

Another way to organize data

  x-axis y-axis

A 10cos60° 10sin60°

B 25cos10° 25sin10°

R    

Adding Vectors by the Component Method

Another way to organize data

  x-axis y-axis

A 5.00 8.66

B 24.62 4.34

R    

Adding Vectors by the Component Method

Another way to organize data

  x-axis y-axis

A 5.00 8.66

B 24.62 4.34

R  29.62  13.00

Adding Vectors by the Component Method

Is this it? 32.33 km?

Ay

Ax

By

Bx

R

Rx

Ry

Rx = 29.60

Ry = 13.00

R = 32.33

Adding Vectors by the Component Method

Is this it? No!! Now we have to find the direction.

Ay

Ax

By

Bx

R

Rx

Ry

Rx = 29.60

Ry = 13.00

R = 32.33

Adding Vectors by the Component Method

tan θ = Ry/Rx

θ = tan-1(Ry/Rx)

R

Rx

Ry

Rx = 29.60

Ry = 13.00

R = 32.33

θ

Adding Vectors by the Component Method

θ = 23.71° N of EHow do we know it is North of East?

R

Rx

Ry

Rx = 29.60

Ry = 13.00

R = 32.33

θ

Adding Vectors by the Component Method

E (east)

N (north)

S (South)

W (west) 40° N of E

A

B

C

Which angle is 10° North of West? A, B or C?Which angle is 10° West of North? A, B or C?What is angle C?

Adding Vectors by the Component Method

A = 10 km @ 60° N of E

B = 25 km @ 10° N of E

Final answer: 32.33 km @23.71° N of E

R

Adding Vectors by the Component Method

Some helpful hints! Never use the original values after the

vector has been resolved. Assign negative values to S and W

components of vectors (assuming N and E are positive)

Always make sure you are in degree mode.

Adding Vectors by the Component Method

Some helpful hints! Always make sure you are in degree

mode. Make sure you draw your vectors in the

correct directions initially. To help, redraw a coordinate system at

the end of each vector. Be organized, stay organized, & finish the

entire problem.

Adding Vectors by the Component Method

Other examples A duck flies 10 m/s @ 30° S of W with a

wind blowing 5 m/s N. What is the resulting velocity of the duck?

Answer:

Adding Vectors by the Component Method

Other examples A duck flies 10 m/s @ 30° S of W with a

wind blowing 5 m/s N. What is the resulting velocity of the duck?

Answer:

A -x & -y

B 0 x & +y

Adding Vectors by the Component Method

Other examples A duck flies 10 m/s @ 30° S of W with a wind

blowing 5 m/s N. Answer:

R = ??

  x-axis y-axis

A -10cos30 -10sin30

B 0 +5

R    

AB

Adding Vectors by the Component Method

Other examples A duck flies 10 m/s @ 30° S of W with a wind

blowing 5 m/s N. Answer:

R = ??

  x-axis y-axis

A -8.66 -5

B 0 +5

R    

AB

Adding Vectors by the Component Method

Other examples A duck flies 10 m/s @ 30° S of W with a wind

blowing 5 m/s N. Answer:

R = ??

  x-axis y-axis

A -8.66 -5

B 0 +5

R -8.66  0 

AB

Adding Vectors by the Component Method

Other examples A duck flies 10 m/s @ 30° S of W with a wind

blowing 5 m/s N. Answer:

R = -8.66 m/s or 8.66 m/s W

  x-axis y-axis

R -8.66  0 

AB

Adding Vectors by the Component Method

ANY and ALL vectors can (and will) be analyzed this way. Displacements Velocities Accelerations Forces Momentum

Adding Vectors by the Component Method

Any questions? These notes will be online You MUST be good at vectors to

succeed / pass this class. Ask questions (in class) whenever

necessary.