1. Starter If the height is 10m and the angle is 30 degrees,
how long is the shadow?
h/s = tanq, so s =h/tanq = 10/tan30 = 17.3m

2. Vectors and Vector Addition
1. Characteristics of Vectors
2. Multiplying a vector by a scalar
3. Adding Vectors Graphically
4. Adding Vectors using Components

3. What is a vector?
A vector is a mathematical quantity with two characteristics:
1. Magnitude or Length
2. Direction ( usually an angle)

4. Vectors vs. Scalars
A vector has a magnitude and direction. Examples: velocity, acceleration, force, torque, etc.

5. Vectors vs. Scalars
A scalar is just a number. Examples: mass, volume, time, temperature, etc.

6. A vector is represented as a ray, or an arrow.
The terminal end
or head V
The initial end or tail

7. Picture of a Vector Named A
Magnitude of A
A = 10
Direction of A
q = 30 degrees

8. The Polar Angle for a Vector
Start at the positive x-axis
and rotate counter-clockwise
until you reach the vector.
That’s how you find the
polar angle.

9. Two vectors A and B are equal if they have the same magnitude and direction.
This property allows us to move vectors around
on our paper/blackboard without changing
their properties.

10. A = -B says that vectors A and B are anti-parallel
A = -B says that vectors A and B are anti-parallel. They have same size but the opposite direction.
A = -B also implies
B = -A B A

11. Graphical Addition of Vectors ( Head –to Tail Addition )
To find C = A + B :
1st Put the tail of B on
the head of A.
2nd Draw the sum vector
with its tail on the
tail of A, and its head
on the head of B.
Example: If C = A+B, draw C.
Here’s Vector C

12. Graphical Addition of Vectors ( Head –to Tail Addition )
To find C = A - B :
1st Put the tail of -B on
the head of A.
2nd Draw the sum vector
with its tail on the
tail of A, and its head
on the head of -B.
Example: If C = A-B, draw C.
Here’s Vector C = A - B

13. Addition of Many VectorsB B D C C R D
Add A,B,C, and D
R = A + B + C + D

14. Multiplication of a Vector by a Number.

15. Vector Addition by Components (Do the math)

16. A vector A in the x-y plane can be represented by its perpendicular components called Ax and Ay.
Components AX and AY
can be positive, negative,
or zero. The quadrant
that vector A lies in
dictates the sign of the
components.
Components are scalars. A AY x AX

17. When the magnitude of vector A is given and its direction specified then its components can be computed easily y
AX = Acosq A AY
AY = Asinq  x AX
You must use the polar angle in these formulas.

18. Example: Find the x and y components of the vector shown if A = 10 and q = 225 degrees.
AX = Acosq = 10 cos(225) =
Ay = Asinq = 10 sin(225)
A = (-7.07, -7.07)

19. The magnitude and polar angle vector can be found by knowing its components
 = tan-1(AY/AX) + C

20. Example: Find A, and q if A = ( -7.07, -7.07)
= 10
 = tan-1(AY/AX) + C
= tan-1(-7.07/-7.07) + 180
= 225 degrees

21. Example: Find A, and q if A = ( 5.00, -4.00)
= 6.40
 = tan-1(AY/AX) + C
= tan-1(-4.00/5.00) + 360
= 321 degrees

22. A vector can be represented by its magnitude and angle,
or its x and y components. You can go back and forth’
from each representation with these formulas:
If you know Ax and Ay
you can get A and q with:
If you know A and q, you can get Ax and Ay with:
Ax = Acosq
Ay = Asinq

23. Adding Vectors by Components
If R = A + B
Then Rx = Ax + Bx
and Ry = Ay + By
So to add vectors, find their components and add the like components.

24. Example
A = ( 3.00,2.00) and B = ( 0, 4.00)
If R = A + B find the magnitude and direction of R.
Solution: R = A + B = ( 3.00,2.00) + ( 0, 4.00),
so R = ( 3.00, 6.00)
Then R = ( )1/2 = 6.70
q = tan-1( 6/3) = 63.4o

25. Example If R = A + B find the magnitude and direction of R.
1st: Find the components of A and B.
Ax = 10cos 30 = 8.66
Ay = 10 sin30 = 5.00
Bx = 8cos 135 = -5.66
By = 8sin 135 = 5.66
2nd: Get Rx and Ry
Rx = Ax + Bx = = 3.00
Ry = Ay + By = = 10.7
3rd: Get R and q : R = ( )1/2 = 11.1
q = tan-1 ( 10.7/3.00) = 74.3o

26. Summary If R = A + B Rx = Ax + Bx Ax = Acosq Ay = Asinq Ry = Ay + By
If you know Ax and Ay
you can get A and q with:
If you know A and q, you can get Ax and Ay with:
Ax = Acosq
Ay = Asinq
If R = A + B Rx = Ax + Bx
Ry = Ay + By

27. Unit Vectors 𝑖 Examples: A = (3,-2,5) = 3i - 2j + 5k
= ( 1, 0, 0) 𝑗 = ( 0, 1, 0 ) 𝑘 = ( 0, 0, 1)
Examples: A = (3,-2,5) = 3i - 2j + 5k
B = (3,0,5) = 3i + 5k

28. The Scalar or Dot Product
Example: A = 3i +4j +5k B = 4i + 2j– 3k
A.B = – 15 = 5

29. The Cross Product A x B = i( Ay Bz - Az By ) - j( Ax Bz - Az Bx )
+ k( Ax By - Ay Bx )

30. Example
A = (3,0,2) B = (1,1,0)
A X B = 𝒊 𝒋 𝒌 = i (0 -2) -j(0-2)+k(3-0)
= -2i +2j +3k = (-2,2,3)

31. Integrals of Vectors 𝑨 𝑑𝑥=𝒊 𝐴 𝑥 dx + j 𝐴 𝑦 𝑑𝑥+𝒌 𝐴 𝑧 𝑑𝑥
Example: If A = 3xi +5x2j , then find 𝑨 𝑑𝑥.
𝑨 𝑑𝑥=𝒊 3𝑥 𝑑𝑥 j 5𝑥2 𝑑𝑥= 3 2 𝑥2𝒊 𝑥3 𝒋

32. Derivatives of Vectors
dA/dx = d (Ax, Ay, Az)/dx
= (dAx/dx, dAy/dy, dAz/dz)
Example: If A = 3ti - 5t2j , then find dA/dt.
dA/dt = 3i -10tj

33. EXIT
If A = 6cos(3t)i +5cos(10t)j ,
then find 𝑨 𝑑𝑡.