Vectors Flashcards
Vector
a quantity that has both magnitude and direction
Scalar
only have magnitude
Length on a vector
//u//
Displacement
the distance from start to finish
Position Vector
displacement from point P to origin O
Vector Addition - Head to tail
Step 1: move the second vector until its tail touches the head of the first vector
Step 2: form the vector joining the tail of the first to the head of the second, this is now a+b
Vector Addition - Parallelogram Rule
Step 1: move the vectors a and b until their tails collide at a common origin
Step 2: complete a parallelogram based on a and b as 2 adjacent sides
Step 3: the vector from opposite corner of the parallelogram to origin is the sum of a and b
cosine Rule
a^2= b^2 + c^2 - 2bcCosA (capital letters angles)
Sine Rule
a/sinA = b/sinB
Zero Vector
- vector that has no magnitude
denoted by 0
Negative of a vector
in the opposite direction
addition of a negative vector to a positive
a+ - a = 0 vector
When to use cosine rule
when you have 2 sides and 1 angle
When to use sine rule
when you have 2 angles and 1 side or 2 sides and no angle
Subtraction of Vectors
u - v = u+(-v)
Vector subtraction Rule: Head to Tail
Step 1: Reverse the sense of v to create
Step 2: move (-V) so that its tail lies at the head of u
Step 3: join the tail of u to the head of (-v) to form u+(-v) = u - v
Vector subtraction Rule: Tail to Tail
Step 1: move v parallel to itself so that its tail touches the tail of u
Step 2: draw the vector from the head of v to the tail of u, this is u-v
When to use addition rules
when you want to know where youll end up
When to use subtraction rule
when you want to know how you got there
Scalar Multiplication -for s greater or equal to 0
the product SU is defined to be a vector in the same direction as u but with the length s times as long
Scalar Multiplication - for s less than 0
su has length s times that of u but has opposite direction
Unit Vectors
has the same direction of u but has the length of one unit ((1/IuI) x u)
Angle Between Vectors
the lesser of the 2 possible angles between u and v
u + v
v+u
(u+V) + w
u + (v + w)
u + 0
0 + u = u
u + (-u)
0
dot product
is a number denoted by u.v
(length of u)(scalar component of V parallel to u)
/u/ /v2/
/u/ /v/ cos0
dot product theta
angle between the vectors (between 0 and 180)
Scalar Components - V2
/v/ cos0 (parallel)
Scalar Components - V1
/v/ sin0 (perpendicular)
Cross Product - length
/UxV/ = (length of u) (scalar component of V perpendicular to u)
/u/ /v/ sin0
Cross Product - direction
perpendicular to both u and v
Cross Product - sense
given by right hand grip rule
coordinates in a 2D shape
x and y
coordinates in a 3d shape
x,y and z
orthonormal basis
OP = u = u1i + u2j +u3k
Vector Addition
if u = u1i + u2j +u3k and v = v1i + v2j +v3k
then u+v = (u1 +v1)i + (u2 + v2)j + (u3+v3)k
Components of zero factor
0i + 0j + 0k
components of -u
u = u1i + u2j +u3k
-u = (-1) (u1i + u2j +u3k)
= - u1i - u2j - u3k
Vector Subtraction
u = u1i + u2j +u3k and v = v1i + v2j +v3k u-v = (u1-v1)i + (u2-v2)j + (u3-v3)k
Scalar Multiplication
su = su1i + su2j +su3k
When to use dot product
for vectors
when to use cross product
for scalars
Dot product of the orthonormal basis vector
i x j = 0
i x i = 1
Dot product of arbitrary vectors in component form
u.v = u1v1 + u2v2 + u3v3
Length of vector in terms of cartesian components
/u/ = square root of u1^2 + u2^2 = u3^2
Angles between two vectors from cartesian component
cos0= u.v/ /u//v/
= (u1v1 +u2v2 + u3v3 )/ (square root of u1^2 + u2^2 + u3^2 ) x (square root of v1^2 + v2^2 + v3^2)
Unit Vector
^u = (1/ /u/)(u1i +u2j + u3k)
Work
Force x length times cos0
Unit vector
Divide vector formula by length
Displacement
Distance from a to b
Work (Cartesian)
Force x displacement
Torque
R x f x sin0
