Linear Algebra
1 Geometric vectors
A vector has two distinct properties: a magnitude, which is a non-negative scalar, and a direction. Thus, two vectors are equal if they have the same magnitude and direction. They may be moved parallel to coincide.
1.1 Addition of geometric vectors
Tip to tail method
Parallelogram law of addition
Commutative law of addition Associative law of addition
1.2 Multiplication by a scalar
The vector lengthens and compresses according to the magnitude of a multiplied scalar and switches direction if the scalar is negative. The zero vector 0 has a length of |0| and points in every direction. It is created by multiplying the scalar zero with a vector, or adding a vector to its negative.
1.3 Subtraction of vectors Vector subtraction
1.4 List of useful properties
Useful properties
1.5 The geometry of parallelograms
A quadrilateral is a figure with four sides. A quadrilateral is a parallelogram if opposite sides are parallel and have the same length. This condition can be proven by checking for simply one pair of opposite sides.
Furthermore, a quadrilateral is a parallelogram if and only if the diagonals bisect each other.
P !v Q
!v
!u
!u +!v
!u +!v
!u
!u
!v
!v
!v +!w =!w+!v (!u +!v) +!w=!u + (!v +!w)
!v !w =!v + ( !w)
The vector sum v + w is represented by the diagonal of the parallelogram formed using sides v and w.
!v +!w =!w+!v
(!u +!v) +!w=!u + (!v +!w)
!v +!0 =!0 +!v =!v ( !v) =!v
!v + ( !v) =!v !v = 0
(µ!v) = ( µ)!v (!v +!w) = !v + !w ( +µ)!v = !v +µ!v 1!v =!v
( )!v = ( !v) ( 1)!v = !v
!v +!w =!w+!v
(!u+!v) +!w =!u+ (!v +!w)
!v +!0 =!0 +!v =!v ( !v) =!v
!v + ( !v) =!v !v = 0
(µ!v) = ( µ)!v (!v +!w) = !v + !w ( +µ)!v = !v +µ!v 1!v =!v
( )!v = ( !v) ( 1)!v = !v
!v +!w =!w+!v
(!u+!v) +!w =!u+ (!v +!w)
!v +!0 =!0 +!v =!v ( !v) =!v
!v + ( !v) =!v !v = 0
(µ!v) = ( µ)!v (!v +!w) = !v + !w ( +µ)!v = !v +µ!v 1!v =!v
( )!v = ( !v) ( 1)!v = !v
Proof 1:
Opposite sides are equal
Proof 2:
Diagonals bisect
2 Position vectors and components
2.1 Magnitude, unit vectors and hat notation Absolute value rule
A unit vector of a vector is denoted with a ‘hat’ and has a length of one. There is always a unit vector pointing in any given direction and thus there are infinitely many unit vectors in the plane or in space.
Unit vector
Proof of unit vector length
Proof of unit vector condition
Division by any non-zero scalar
2.2 Parallel vectors
Parallel vectors point in either the same or opposite directions. The zero vector is thus parallel to every vector.
Non-zero vectors are parallel if and only if they are scalar multiples of each other.
Parallel condition
Vectors point in the same direction
Vectors point in opposite directions
3
| !v|=| ||!v|
!ˆv = 1
|!v|!v = !v
|!v|
!v P roof:|!ˆv|=|1
|!v|
!v|=| 1
|!v|||!v|= 1
|!v||!v|= 1
P roof :|!v|!ˆv =|!v|(1
|!v|
!v) = (|!v| 1
|!v|)!v = 1!v =!v
!v = !w We assume that PQ = SR and we need to show that QR = PS.
P
R Q
S
QR!=QP!+P S!+SR!
= P Q!+P S!+P Q!
=P Q! P Q!+P S!
=0+P S!
=P S!
P
R Q
S
T
P Q!=P T!+T Q!
=T R!+ST!
=ST!+T R!
=SR!
If vectors point in the same direction,
!ˆv = ˆ!w
so that!v =|!v|!ˆv =|!v|!wˆ =
|!v|
|w!|
!w
!ˆv = ˆ!w
If vectors point in opposite directions,
!ˆv = !wˆ
so that!v =|!v||!ˆv|=|!v|( !wˆ) =
|v|!
|!w|
!w
!ˆv = !wˆ
2.3 Position vectors and components
Every geometric vector decomposes in terms of i, j and k, the unit vectors in the x-, y- and z- directions respectively. k is signified by (0, 0, 1) and so forth.
The position vector of a point P in the plane or in space is the vector that is the directed segment of a line emanating from the origin and terminating at P.
Position vector
The cartesian form of a vector describes the decomposition of the vector with respect to i, j and k, where a, b and c are length coordinates. To add, subtract and negate vectors, perform the operation on each component. To multiply a vector by a scalar, multiply each component by the scalar.
Cartesian form
Vector components
2.4 Length of a vector
Vector length
2.5 Linear independence of two vectors
Two vectors are linearly independent if and only if they are not parallel.
Linear independence
3 Dot products and projections
3.1 Geometric definition of the dot product Geometric dot product
Projections
!v = ! OP
P(a, b) :!v =a!i +b!j P(a, b, c) :!v =a!i +b!j +c!
k
!v = !
P Qhas components
!v = (a2 a1)!i + (b2 b1)!j + (c2 c1)!k
If!v =a!i +b!j , then|!v|=p a2+b2 If!v =a!i +b!j +c!
k , then|!v|=p
a2+b2+c2
a!x +b!y =!0
!v ·!w =|!v||!w|cos✓
!v ·!w
|!w| =|!v|cos✓
!v ·!w
|!v| =|!w|cos✓
!w
!v
✓
!w
!v
✓
Cauchy-Schwarz inequality
Proof
Orthogonality criterion
3.2 Algebraic definition of the dot product Algebraic dot product
Consequences
Proof of the distributivity of dot over plus
The cosine rule
3.3 The angle between two vectors
The angle between two vectors
3.4 Projections and orthogonal components
A vector v can be thought of as a composition of two vectors, one parallel to w and one perpendicular (orthogonal) to w. Therefore, we firstly look for a vector projection of v in the direction of w to find the ‘best approximation’ of v using a scalar multiple of w, as well as secondly, the real scalar component of v in the direction of w, and thirdly, the vector component of v orthogonal to w.
5
|!v ·!w|6|!v||!w|
If v and w are mutually perpendicular,
!v ·!w = 0
!v ·!w =ad+be+cf
cos✓= !v ·!w
|!v||!w|
(1) ✓is acute , 0<!v ·!w <|!v||!w| (2) ✓is obtuse , |!v||!w|<!v ·!w <0 (3) ✓= ⇡
2 , !v ·!w = 0
!v
!w
✓
|!v|cos✓ P
O Q
(1) !
OQ= (|!v|cos✓) ˆ!w = !v ·!w
|!w|2 !w
(3) !
QP =!v !v ·!w
|!w|2 !w (2)|!v|cos✓= !v ·!w
|!w| 1cos✓1
|!v||!w||!v||!w|cos✓|!v||!w|
|!v||!w| !v ·!w |!v||!w|
!u ·!v =!v ·!u
(!u +!v)·!w =!u ·!w+!v ·!w
( !u)·!v = (!u ·!v)
!v ·!v =|!v|2 so |!v|=p!v ·!v
!u ·!v =!v ·!u
(!u +!v)·!w =!u ·!w +!v ·!w
( !u)·!v = (!u ·!v)
!v ·!v =|!v|2 so|!v|=p!v ·!v
!u ·!v =!v ·!u
(!u +!v)·!w =!u ·!w +!v ·!w
( !u)·!v = (!u ·!v)
!v ·!v =|!v|2 so|!v|=p!v ·!v
!u ·!v =!v ·!u
(!u +!v)·!w =!u ·!w +!v ·!w
( !u)·!v = (!u ·!v)
!v ·!v =|!v|2 so|!v|=p!v ·!v
(!u +!v)·!w = ((u1+v1)i+ (u2+v2)j+ (u3+v3)k)·!w
= (u1+v1)w1+ (u2+v2)w2+ (u3+v3)w3
=u1w2+v1w1+u2w2+v2w2+u3w3+v3w3
= (u1w1+u2w2+u3w3) + (v1w1+v2w2+v3w3)
=!u ·!w +!v ·!w
z2=x2+y2 2xycos✓
4 Cross Products
4.1 Definition of the cross product
The cross product always produces a vector perpendicular to the original two vectors. In other words, v x w is always perpendicular to both v and w.
Algebraic cross product Matrix method:
4.2 List of useful properties
Useful properties
4.3 Method of expanding brackets
The properties listed in the previous section yield another method for finding cross products, namely, expanding the brackets and using the facts about i, j and k to obtain the result. i x i = j x j = k x k = 0, i x j = k, i x k = -j, j x
k = i, j x i = -k, k x i = j, k x j = -i.
4.4 Geometric interpretation Geometric cross product
One should measure the angle between the two vectors to lie between 0-180 degrees, so that the result is positive.
Parallelogram area
Triangle area
The Right-Hand Rule
i j k i j
v1 v2 v3 v1 v2
w1 w2 w3 w1 w2
!v ⇥ !w = (v2w3 v3w2)!i + (v3w1 v1w3)!j + (v1w2 v2w1)! k
)
i j k i j
v1 v2 v3 v1 v2
w1 w2 w3 w1 w2
– – + + +
)
Collect the down products first, e.g. 12i, 12j, 5k, -8k, -15i, -6j to get the cross product; add each of these items together for the result
|!v ⇥ !w|
|!v ⇥ !w| 2
|!v ⇥ !w|=|!v||!w|sin✓ u
!v !w
!u
!v ⇥ !w is a vector
(!v ⇥ !w)·!v = (!v ⇥ !w)·!w = 0
!v ⇥ !w = (!w ⇥ !v)
!v ⇥ !v =!0
( !v)⇥ !w = (!v ⇥ !w) =!v ⇥( !w) (!u +!v)⇥ !w =!u ⇥ !w+!v ⇥ !w
!u ⇥(!v +!w) =!u ⇥ !v +!u ⇥ !w
!v ⇥ !w is a vector
(!v ⇥ !w)·!v = (!v ⇥ !w)·!w = 0
!v ⇥ !w = (!w⇥ !v)
!v ⇥ !v =!0
( !v)⇥ !w = (!v ⇥ !w) =!v ⇥( !w) (!u +!v)⇥ !w =!u ⇥ !w+!v ⇥ !w
!u ⇥(!v +!w) =!u ⇥ !v +!u ⇥ !w
The cross product must lie perpendicular to both multiplied vectors.
If the fingers of your right hand are curled in the direction of rotation of v until it points in the direction of w, then the outstretched thumb points in the direction of the cross product direction u.
