Vectors Flashcards
Column Vectors
A translation can be described using column vectors.
A column vector describes the horizontal movement (x) and vertical movement (y) these are vector components.
Vectors are indicated with a wavy line underneath the letter of AB with an arrow pointing in the direction of the vector
Note
Vectors can start anywhere on the number plane
Multiplying a vector by a scalar
If s is any number (scalar) and a is any vector then sa is another vector
If a = (x|y) then sa = (xs|ys)
Note s is called a scalar because it usually changes the scale of the vector
Inverse
The inverse of a vector is obtained by changing the sign of the components of the vector
Note
This has the effect of changing the direction of the arrow on the vector
Addition and subtraction
Vectors can be added or subtracted by performing one translation after another
If a = (x|y) and b = (p|q) then a + b = (x+p | y+q)
Note
a + b = b + a this is called the commutative rule for addition of vectors
If a = (x|y) and b = (p|q) then a – b = (x–p, y–q)
Note
Vector subtraction is best done by addition of the inverse or opposite vector
Magnitude or modulus of a vector
The magnitude or size of a vector is represented by its length, the longer the length the greater the magnitude
The magnitude of a~ or AB→ is denoted by |a~| or |AB→| respectively and calculated using Pythagoras’ theorem
a = (x|y)
Magnitude of vector a |a| = √(x² + y²)
Argument (angle) of a vector θ
The argument of a~ or AB→ is denoted by θ and is calculated using Pythagoras’ theorem
a = (x|y)
Argument of vector a~
tanθ = y/x
θ = tan⁻¹(y/x)
Vectors in 3 dimensions
The usual convention is to take x and y axis in a horizontal plane. Now in 3 dimensions we add z axis pointing vertically upwards
Magnitude or modulus of a vector in 3D
a = (x,y,z)
|a| = √(x² + y² + z²)
Displacement and position vectors
Displacement can start from anywhere
Position vectors are displacement vectors that start at the origin
Parallelogram rule of addition
To identify the point with the position vector OA> + OB> is not easy because the arrows from O to A and from O to B are not related in the way needed for addition
Therefore the parallelogram OACB must be completed
OA> + OB = OA> + AC> = OC>
Note
OB> = AC>
Parallelogram rule of subtraction
To identify the point with position vector OB> – OC> is not easy because the arrows from O to A and from O to B are not related in the way needed for subtraction
Note
Vector subtraction is best done by addition of the inverse
Therefore the parallelogram O(–A)CB must be completed
OB> – OA> = OB> + BC> = OC> = AB>
Note
OA> = –OA> = BC> and OC> = AB>
Therefore
AB> = OB> – OA>
Component vectors
Apply the rules of vector algebra to a vector in column vector form,
The vector (1|0) and (0|1) are called unit vectors in the x and y directions.
They are denoted by the letter i = (1|0) and j = (0|1)
Component vectors in 3D
a = (x|y|z) and b (q|r|s) are written in component form a.b = xq + yr + zs
where i = (1|0|0) , j = (0|1|0) and
k = (0|0|1)
Dot products (scalar products) of 2 vectors
If you have 2 vectors which are not identical but have equal magnitude, then you can write |a~| = |b~|
If s is a scalar multiple of a~ then it follows from the definition that sa = |s|.|a|
Note
This is true when s is positive, negative or zero
The dot products of vector a~ and b~ is a number a.b = |a|.|b|.cosθ, where θ is the angle between the directions of a and b.
Note
The angle θ may be acute or obtuse but it is important that it is the angle between a and b not the angle between a and –b
Note
For unit vectors i, j and k ii = jj = kk = 1 and ij = jk = ki = 0
Dot products (scalar products) in component form (2D)
a = (x|y) and b = (p|q) are written in component form a.b = xp + yq
Dot products (scalar products) in component form (3D)
a = (x|y|z) and b = (p|q|r) are written in component form a.b = xp + yq + zr
where i = (1|0|0) , j = (0|1|0) and
k = (0|0|1)
Properties of dot products
s . (a~.b~) = (sa~) . b~
If s > 0 then the angle between sa~ and b~ is θ
If s < 0 then the angle between sa~ and b~ is π – θ
Commutative rule for scalar products
p~.q~ = q~.p~
Distributive rule for scalar products
(p~ + q~) . r = p~.r + q~.r
Perpendicular (orthogonal) vectors
If vectors a~ and b~ are perpendicular then a.b = 0
If neither a nor b is a zero vector (0|0) and a.b = 0 then the vectors are perpendicular
Parallel vectors
If vectors a~ and b~ are parallel then a~.b~ = ± |a~||b~|
Angle between 2 vectors
cosθ = a~.b~ / [|a|.|b|
a.b = |a~|.|b~|.cosθ
x₁x₂ + y₁y₂ / [√(x₁)²+(y₁)² . √(x₂)²+(y₂)²
Steps:
1. Modulus of a
2. Modulus of b
3. a.b
4. solve for θ
Unit vector of a~
A unit vector with magnitude 1 like i, j and k. To find the unit vector in the same direction as a given vector, divide the vector by its magnitude
Unit of vector a~
a~/|a~| = x₁i + y₁j / [√(x₁)²+(y₁)²
Vector equation of a line in 2D
A vector equation of a line passing through a fixed point 𝐴 with position vector 𝑎 and parallel to a vector 𝑏 is 𝑟 = 𝑎 + 𝑡𝑏, where 𝑡 is a scalar parameter. This is called the vector equation of the line.
Note
To find the vector equation of a line, you need to have either
* A point it goes through and its direction or
* Two points it goes through.
Pairs of lines in 2D
In 2D a pair of lines are either parallel or they intersect
The lines with vector equation r = a + sp and r = b + tq have the same direction if p is a multiple of q. If in addition b - a or a - b is a multiple of q then lines are the same, otherwise the lines are parallel
Note:
2 equations may represent the same line even though the vector a, b, p, q are different
Vector equation of a line in 3D
The same principles involved in 2D vectors carries on into 3D vectors except for gradient (direction is used instead) and in 3D non-parallel lines may or may not meet.
If they don’t meet they are skew.
The lines with vector equation r = a + sp and r = b + tq intersect if unique values of s and t can be found such that a + sp = b + tq. If 2 unique values can not be found then the lines are skew
A vector equation of the line through 2 fixed points A and B with position vectors a and b is given by the equation r = a + t(b – a) where t is a scalar parameter.
Distance from a point to a line
r = a + tp the vector p can be any vector in the direction of the line, in the case when p is a unit vector u, the equation becomes r = a + tu.
In this case |t| is the distance along the line from A to the point with parameter t.
Note: this principal is the same in 3D
