VECTOR ALGEBRA
▪ Magnitude – A numerical value with appropriate units.
▪ Scalar is a quantity that is completely specified by magnitude.
▪ Vector requires both, magnitude and direction for a complete description.
The main difference between scalars and vectors is difference between scalar and vector algebra
– geometrical description of vector quantity:
▪ the direction of a vector is the counterclockwise angle of rotation
which that vector makes with x - axis.
▪ Two vectors are equal if they have the same magnitude and directions.
▪ Vectors that have the same magnitude and the same direction are the same.
▪ Multiplying a vector by a scalar: Multiplying vector by 2 increases its magnitude by a factor 2.
▪ Opposite vectors: 𝐴⃗ 𝑎𝑛𝑑 − 𝐴⃗
Multiplying by –½ changes the magnitude ½ times and reverses the direction
▪ Components of a Vector (x or horizontal component and y or vertical component)
Any vector can be “resolved” into two components.
 If you know the magnitude A and direction θ of a vector
if (A,θ) 
known
A
= (A, θ) you can find x and y components of that vector:
A x = Acosθ
v = 34 m/s @ 48° . Find vx and vy
A y = Asinθ
vx = (34 m/s )cos 48 ) = 23 m/s;
vy = (34 m/s) sin 48° = 25 m/s
 If you know x and y components of a vector
A
you can find the magnitude A and direction θ of that vector:
Fx = 4 N and Fy = 3 N .
A= A2x +A2y
if (A x ,A y ) 
known
 A  if the vector is in the first quadrant;
θ=arc tan  y  if not you find it from the picture.
 Ax 
Find magnitude (always positive!) and direction.
F = 42 +32 = 5N ;
 = arc tan (¾) = 370
▪ Vector addition graphically – comparison between “head-to-tail” and “parallelogram” method
Two methods for vector addition are equivalent.
examples: Find 𝑣⃗1 + 𝑣⃗2 = 𝑣⃗ and 𝐹⃗1 + 𝐹⃗2 = 𝐹⃗ using both methods
"head-to-tail" method
parallelogram method
Sum is known as the "resultant" or “net vector”
The resultant is the vector sum of the two individual
vectors.
examples: resultant velocity,
resultant force or net force
(the one that can replace all forces
that are applied at the same time)
 SUBTRACTION is adding opposite vector.
 
C = A - B = A + -B
▪ ANALYTICALLY:
⃗⃗
𝐶⃗ = 𝐴⃗ + 𝐵
𝐶𝑥 = 𝐴𝑥 + 𝐵𝑥 = 𝐴 cos 𝜃𝐴 + 𝐵𝑐𝑜𝑠 𝜃𝐵
𝐶𝑦 = 𝐴𝑦 + 𝐵𝑦 = 𝐴 𝑠𝑖𝑛 𝜃𝐴 + 𝐵𝑠𝑖𝑛 𝜃𝐵
𝐶 = √𝐶𝑥2 + 𝐶𝑦2 ;  from the picture
F = 68 N @ 24°
1
F2 = 32 N @ 65°
Find
F  F1  F2
Fx = F1x + F2x = 68 cos240 + 32 cos650 = 75.6 N
Fy = F1y + F2y = 68 sin240 + 32 sin650 = 56.7 N
F  Fx2  Fy2  94.5 N
 = arc tan (56.7/75.6) = 36.90
F  94.5N @ 370