Vector
Scalar
Equal Vectors
A quantity that requires both
magnitude and direction.
A quantity that requires only
magnitude.
Two vectors are equal if they have the
exact same magnitude and exact
same direction.
Ex) Velocity (not speed!)
Ex) Speed (not velocity!)

Tip/Head
 You can multiply a scalar by a
vector.
 You CANNOT divide a vector by a
scalar – you must multiply it by its
reciprocal.
Tail
Set the components in each
individual vector equal to each
other.

𝑥
𝑎
for [ ] = [𝑦]
𝑏
𝑎 = 𝑥 and 𝑏 = 𝑦
Particularly useful when dealing with
shapes or comparing vectors.
Negative Vectors
Zero Vector
Magnitude
Negative vectors have the same
length, but are perfectly opposite in
direction.
A vector that has a magnitude of 0
and no defined direction.
The length of a vector. Also called
“modulus”. Represented as |𝑎|.
Defined as ⃗0.
𝑥
If 𝑎 = [𝑦] ,
𝑧
𝑎 + (−𝑎) = ⃗0
𝑣
|𝑎| = √𝑥 2 + 𝑦 2 + 𝑧 2
−𝑣

|𝑎| is always positive
 |𝑎| is a scalar
Helps you find lengths and
distances – useful with shapes
(triangles).
Vector between two points
Addition of Vectors
Component form of a vector
To find the vector between two
points, subtract the first from the
second:
Geometric: Add the vectors tip-to-tail.
For algebraic vectors rather than
geometric vectors.
For points 𝐴(𝑎1 , 𝑎2 , 𝑎3 )
and 𝐵(𝑏1 , 𝑏2 , 𝑏3 ),
𝑏1 − 𝑎1
⃗⃗⃗⃗⃗ = [𝑏2 − 𝑎2 ]
Vector 𝐴𝐵
𝑏3 − 𝑎3
Don’t forget to draw the actual
answer (the resultant – from the tail
of the first to tip of the last)
𝑢
⃗
𝑣
𝑣
𝑢
⃗
Algebraic: Add the individual
components.
𝑎1 + 𝑏1
⃗
𝑎
𝑎 + 𝑏 = [ 2 + 𝑏2 ]
𝑎3 + 𝑏3
A vector can be written in its
𝑥
“component” form where 𝑎 = [𝑦]
𝑧
where 𝑥, 𝑦 and 𝑧 represent the
amount of units in each direction on
the axis.
Ex) 𝑎 = [
2
]
−1
Right 2
Down 1
Standard Basis Vectors/
Unit Vector Form
All vectors can be created from the
vectors:
1
0
0
⃗ = [0]
𝑖 = [0], 𝑗 = [1], 𝑘
0
0
1
1
0
(or 𝑖 = [ ], 𝑗 = [ ], in 2D).
0
1
Used regularly with vector addition, scalar
multiplication, dot product, properties
etc…. it is just as common to see 3𝑖 +
3
⃗ as [ 4 ]
4𝑗 − 𝑘
−1
Parallel Vectors
Collinear Vectors
If 𝑎 is parallel to 𝑏⃗, then there exists a
scalar k such that 𝑎 = 𝑘𝑏⃗.
Three or more points are said to be
collinear if they lie on the same
straight line.
In general, 𝑘𝑎 is a vector parallel to 𝑎,
where 𝑘𝜖ℝ.
 It has the same direction as 𝑎,
if 𝑘 > 0 , and
 It has the opposite direction
as 𝑎, if 𝑘 < 0 .

ka  k a
Points 𝐴, 𝐵 and 𝐶 are collinear if ⃗⃗⃗⃗⃗
𝐴𝐵 =
⃗⃗⃗⃗⃗
𝑘𝐵𝐶 for some scalar k.
Used when working with shapes
(showing three points do not form a
triangle for example but that instead
the points all lie on the same line. )
Useful in many applications, shapes,
scalar multiples etc….
Useful when working with ratios of
division.
** See Vector Notation Card**
Vector Subtraction
Scalar Multiplication
Dot Product
Vectors are not subtracted…rather
you add the opposite vectors.
Geometrically this means that you
change the direction and then attach
the vectors tip-to-tail.
You can multiply any vector by a
scalar 𝑘 to change its length or switch
it to the opposite direction.
Used to help find angles between
vectors (tail-to-tail). Particularly
important with perpendicularity.
Note: There is no such thing as dividing a
vector by a scalar; you must multiply it by
⃗ ∙𝒘
⃗⃗⃗ = 𝑣1 𝑤1 + 𝑣2 𝑤2 + 𝑣3 𝑤3
𝒗
1
𝑎⃗
3
3
the reciprocal. i.e. 𝑎, NOT
In general, 𝑘𝑎 :
 has the same direction as 𝑎, if 𝑘 > 0 ,
and
 has the opposite direction as 𝑎, if 𝑘 < 0.
(Algebraic)
⃗ ∙𝒘
⃗ ||𝒘
⃗⃗⃗ = |𝒗
⃗⃗⃗ |𝐜𝐨𝐬𝜽
𝒗
(Geometric)



If 0 ≤ 𝜃 < 90°, cos 𝜃 > 0 so 𝑣 ∙ 𝑤
⃗⃗ > 0
If 𝜃 = 90°, cos 𝜃 = 0 so 𝑣 ∙ 𝑤
⃗⃗ = 0
If 90° < 𝜃 ≤ 180°, cos 𝜃 < 0 so 𝑣 ∙ 𝑤
⃗⃗ < 0
Useful in almost any and every
application of vectors.
Unit Vector
Coordinate Axes
Dot Product Properties
Useful for simplifying vector
equations. Be very careful of notation
and of what is a scalar vs. a vector!
A unit vector is any vector which has a
magnitude of 1. A unit vector in the
direction of 𝑣 is given by:
𝑎 ∙ 𝑏⃗ = 𝑏⃗ ∙ 𝑎
1
𝑣̂ = |𝑣⃗| 𝑣
(Commutative)
where ̂ indicates a unit vector.
𝑎 ∙ (𝑏⃗ + 𝑐) = 𝑎 ∙ 𝑏⃗ + 𝑎 ∙ 𝑐
(Distributive)
Paricularly useful when needing to
change or find sizes of certain vectors
and also has certain properties when
using “Dot Product Properties” (see
card).
Hint: If you have three points in 3D
that form a shape, it is easier to just
draw a sketch in 2D rather than try to
do it in 3D. The result with the math
will be the same!
𝑎 ∙ 𝑎 = |𝑎|2
(Magnitudes Property)
𝑎 ∙ 𝑏⃗ = 0 if 𝑎 ⊥ 𝑏⃗
(Perpendicularity)
Position Vector
Distance
Mid-Point
A vector that has its tail at the origin
and its head at any other point.
Used to find the distance between the
two points and equivalently the
distance of a vector from point 𝐴 to 𝐵
⃗⃗⃗⃗⃗ ).
(aka 𝐴𝐵
Mid-point:
All vectors can be represented as
position vectors.
If you see an "𝑂" as part of the
vector, this
indicates
⃗⃗⃗⃗⃗
origin. i.e. 𝑂𝑃
to the right.
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2


Always positive!
Closely related to the magnitude
or modulus of a vector.
𝑥1 +𝑦1
2
(
,
𝑥2 +𝑦2
2
,
𝑧1 +𝑧2
2
)
Use for questions that state “half
way”, “bisector”, “middle” etc….
The midpoint for 2-Space (2D) would
not have the z-coordinate.
Perpendicular Vectors
Vector Notation
Tip-to-Tail
Two vectors are perpendicular if 𝑎 ∙
𝑏⃗ = 0.
Must be used with vectors!
You use: 𝑎
Text/IB: 𝒂
Used when adding vectors. Can be
used for more than just two vectors.
Notice that if you see tip-to-tip, one of
these two vectors will be the
resultant!
Useful for geometric shapes (right
angle triangles, rectangles) and slopes
of lines (negative reciprocal).
⃗
Be careful with the zero vector! 0
Tip-to-Tip
– Means
one of
these is
the
resultant!
If you see 𝑎, this is just a scalar!
3𝑖
3
⃗ = [ 4 ], NOT [ 4𝑗 ].
Note that 3𝑖 + 4𝑗 − 𝑘
−1
⃗
−𝑘
(You cannot have vectors within vectors!)
Tail-to-Tail
Angle Between Two Vectors
A Vector in the Direction of…
Set two vectors tail-to-tail if you want
to find the angle between the two
vectors.
To find the angle between two
vectors, they must be placed TAIL-toTAIL!
You can create a vector that is in the
same direction as another vector with
whatever length you want by:
See “Dot Product” card for more
details.
1
𝑏⃗ = ±𝑘 ( ) 𝑎
|𝑎|
𝑢
⃗
𝑢
⃗
𝑣
where 𝑏⃗ is the vector that you want to
create and 𝑎 is the vector you have.
𝑣
Notice that you first turn 𝑎 into a unit
vector and then you multiply it by
whatever scalar 𝑘 you would like.