Vector Algebra
Unit vector in k coordinate direction
(0,0,1)
Unit vector in j coordinate direction
(0,1,0)
Unit vector in i coordinate direction
(1,0,0)
Area of a triangle using vector product
(1/2) | a*b |
Associative law for vectors
(a + b) + c = a + (b + c)
Anti-commutative law (vector product)
(a*b) = -(b*a)
Equation of line in component form
(x, y, z) = (1-t)(a1, a2, a3) + t(b1, b2, b3) or t = ((x - a1)/(b1 - a1)) = ((y - b2)/(b2 - a2)) = ((z - b3)/(b3 - a3))
k*j =
-i
i*k =
-j
j*i =
-k
If 3 vectors are co-planar, the triple scalar product =
0
If any 2 vectors are parallel, the triple scalar product =
0
i*i =
0
i.j =
0
j*j =
0
j.k =
0
k*k =
0
k.i =
0
i^2 =
1
j^2 =
1
k^2 =
1
The 2 things required to calculate the direction cosines (Cartesian system)
1. Coordinates of the point 2. The length from the origin to the point
Cartesian vector method for solving resultant
1. Express components as Cartesian vectors 2. Sum the vectors
3 Examples of vectors
1. Force 2. Velocity 3. Acceleration
3 Examples of scalars
1. Length 2. Area 3. Volume 4. Mass 5. Time
2 Ways vectors are used
1. Locate points in 3-D 2. Describe points, lines and planes in 3-D
2 Features of a Dyad
1. Magnitude 2. 2 Directions 3. 9 Components
2 Features of a Triad
1. Magnitude 2. 3 Directions 3. 27 Components
2 Other words for vector magnitude
1. Modulus 2. Length
3 Types of vectors
1. Position 2. Line 3. Free
3 Points about a scalar product
1. Produces a number 2. Product of vectors 3. Essential to use dot
Scalar method for solving resultant
1. Resolve components into x, y and z components 2. Sum components algebraically 3. Find resultants across each axis
3 Points about vector product
1. The 3 vectors form an orthogonal right-handed set 2. Creates a vector 3. Order of multiplication matters (b*a opposite to a*b)
The 2 triple product of vectors
1. Triple vector product: (a*b)*c 2. Triple scalar product: (a*b).c
The 2 types of vector multiplication
1. Vector/cross product 2. Scalar/dot/inner product
Determinant method of vector product
1. Write unit vector and 2 vectors as 3 matrix rows 2. Solve as a determinant of the matrix
Anti-parallel vectors definition
2 Vectors with the opposite direction, and different magnitude eg a = -2b
Parallel vectors definition
2 Vectors with the same direction, but different magnitude eg a = 2b
Any free can be represented by ...
3 Coordinates
In printed work vectors are shown as ...
A bold interface
Principle of transmissibility
A force may be applied anywhere along its line of action and still create the same moment about the origin (M = F*r1 = F*r2 = F*r3)
3 Distinct points that do not lie on the same line form ....
A unique plane
A set of numbers or 3-tuple represent ...
A vector in 3D
Scalar multiplication produces ...
A vector parallel/anti-parallel to the original vector
Line vector definition
A vector that can 'slide' along its line of action
Position vector definition
A vector that gives the position of a point relative to a fixed point
Free vector definition
A vector that is completed defined by its magnitude and direction
Unit vector definition
A vector whose modulus is 1
Position coordinates cannot be ...
Added or subtracted
Work done definition
Amount of displacement along the direction of the applied force
In handwritten work vectors are usually shown as ...
An underlined bold interface
Vectors cannot be ... like scalars
Cancelled
Associative law (scalar product)
Cannot have a scalar product of 3 vectors (a.b.c is not defined)
Within the Cartesian system, the position of any point is given by ...
Coordinates (x,y,z) or components (x1,x2,x3)
The sum a + b of vectors is represented by the ....
Diagonal in the parallelogram rule
When 2 distinct points are known you can ...
Draw a unique straight line through them
Triangle law for vectors
If 2 vectors are represented in magnitude and direction by 2 sides of a triangle in order, their sum is represented by the closing 3rd side
Polygon law for vectors
If lines are drawn to represent 5 vectors, the closing side represents the summed vector
Vectors can be represented geometrically by ...
Line segments in space
Vectors are represented by ...
Lines with arrows
Moment of a force
M = r*F = | F | | r | sin(θ) n ̂ F = force magnitude r = perpendicular distance from origin to force line of action θ = angle between line of action and axis n ̂ = unit vector normal to r and F
The length of a vector line represents its ...
Magnitude
Vectors have ...
Magnitude and direction
Scalars have ...
Magnitude only
Cartesian coordinates definition
Numbers that indicate a point location fixed to the origin, in 2-D or 3-D axes at right angles
A normal vector specifies the ... on a plane
Orientation
Moments add by the ....
Parallelogram rule
A line on a plane is ... to the normal of the plane
Perpendicular
A vector must satisfy the ....
Rules of addition and multiplication
Direction cosines (Cartesian system) definition
The cosines of the angles that OP makes with each axis OP = a line from the origin to a point
Parallelogram rule for vectors
The sum/resultant of 2 vectors is found by forming a parallelogram with the 2 vectors as adjacent sides
The subtraction law for vectors shows ...
The vector of a line joining 2 points is the difference of the point position vectors
The unit vector n ̂ changes direction when ...
The vector product is reversed
The triple scalar product can be used to calculate ...
The volume of a parallelogram
If 2 vectors are perpendicular ...
Their scalar product is 0
If 2 vectors are parallel ....
Their vector product is 0
2 Vectors are equal if ...
They have the same magnitude and direction
A normal vector and the coordinates of one point on the plane ....
Uniquely identify a specific plane
(a*b).c = a.(b*c) = b.(c*a) all give the same ... for a parallelogram
Volume
Vector product equation
a * b = | a | | b | sin θ n ̂ θ = angle between vectors n ̂ = unit vector normal to vectors
Commutative law for vectors
a + b = b + a (order not important)
Subtraction law for vectors
a - b = a + (-b)
Distributive law (vector addition, vector product)
a*(b + c) = (a*b) + (a*c)
Non-associative law (vector product)
a*(b*c) ≠ (a*b)*c
Distributive law (vector product)
a*(λb) = (aλ)*b
Vector product equation (component form)
a*b = (a2*b3 - a3*b2)i - (a1*b3 - a3*b1)j + (a1*b2 - a2*b1)k
Distributive law (vector addition, scalar product)
a.(b + c) = a.b + a.c
Distributive law (scalar product)
a.(λb) = (aλ).b = λ(a.b)
Scalar product equation (component form)
a.b = a1b1 + a2b2 + a3b3
Commutative law (scalar product)
a.b = b.a | a | | b | cos θ = | b | | a | cos θ
Scalar product equation (geometrical form)
a.b = | a | | b | cos θ θ = angle between vectors
a.a =
a1^2 + a2^2 + a3^2
Component form of a vector
ai + aj + ak
Cosine rule
a² = b² + c² - 2bcCosA
j*k =
i
k*i =
j
i*j =
k
Distributive law for vectors
k(a + b) = ka + kb k = scalar value
Direction cosines (Cartesian system) equations
l = direction cosine in x axis m = direction cosine in y axis n = direction cosine in z axis x,y,z = coordinates r = length of line from origin to point
Cartesian equation of a plane
n1*x + n2*y + n3*z = p
Equation of line in vector form
r = a + t(b - a) or r = (1 - t)a + tb a = position vector of line b = position vector of line t = multiple of length ab
Any vector can be written in components of the unit vectors as ...
r = xi + yj + zk
Example of a Cartesian vector
r = xi + yj + zk
Vector equation of a plane
r.n = a.n or r.n = p r = line on plane n = normal vector a= position vector of point in plane p = plane
If vector analysis is used, the moment of a force about an axis =
u.(F*r) u = dreciton of axis r = distance to any point on line of action of force F = force
Moment of a force (magnitude)
| M | = | F | | r | sin θ | F | = force magnitude | r | = perpendicular distance from origin to force line of action θ = angle between line of action and axis
Area of a parallelogram using vector product
| a*b |