Vectors and bases #rvv
A vector is an arrow with a length and a direction. Just like positions, vectors exist before we measure or describe them. Unlike positions, vectors can mean many different things, such as position vectors, velocities, etc. Vectors are not anchored to particular positions in space, so we can slide a vector around and locate it at any position.
Change:
Two vectors, which may or may not be the same vector. Moving a vector around does not change it: it is still the same vector. #rvv‑fc
Notation note
Some textbooks differentiate between free vectors, which are free to slide around, and bound vectors, which are anchored in space. We will only use free vectors.
Notation note
We will use the over-arrow notation →a for vector quantities. Other common notations include bold \boldsymbol{a} and under-bars \underline{a}. For unit (length one) vectors we will use an over-hat \hat{a}.
Vector bases #rvv‑sv
To describe vectors mathematically, we write them as a combination of basis vectors. An orthonormal basis is a set of two (in 2D) or three (in 3D) basis vectors which are orthogonal (have 90° angles between them) and normal (have length equal to one). We will not be using non-orthogonal or non-normal bases.
Any other vector can be written as a linear combination of the basis vectors:
Components of a vector.#rvv‑ec
\vec{a} = a_1 \,\hat{\imath} + a_2 \,\hat{\jmath} + a_3 \,\hat{k}
The numbers a_1, a_2, a_3 are called the components of \vec{a} in the \,\hat{\imath}, \hat{\jmath}, \hat{k} basis. If we are in 2D then we will only have two components for a vector.
Writing a vector as the sum of scaled basis vectors. The scale factors are the components of the vector. Here \vec{a} = 3\hat\imath + 2\hat\jmath, so the components of \vec{a} are a_1 = 3 and a_2 = 2. #rvv‑fb
We draw the symbol \odot (arrow tip) to indicate a vector coming out of the page, and \otimes (arrow fletching) to indicate an arrow going into the page.
Two standard arrangements of the basis vectors when working in 2D. Either \hat\jmath is the vertical and \hat{k} is out of the page, or \hat{k} is the vertical and \hat\jmath is into the page. In both cases \hat\imath is horizontal. #rvv‑f3
Notation note#rvv‑ic
Just as for position coordinates, we can write the vector components 3\hat\imath + 2\hat\jmath as the ordered list (3, 2) if we know which basis we are using. Because we often will be using several bases simultaneously, we will generally write the components explicitly in the 3\hat\imath + 2\hat\jmath form.
Did you know?#rvv‑ii
The use of the letter i,j,k for basis vectors is due to William Hamilton, who was motivated by thinking of basis vectors as extensions of the complex number i. This notation was popularized by the book Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics Founded upon the Lectures of J. Willard Gibbs (1901), by E. B. Wilson. This book also introduced the use of bold letters to represent vectors.
Length of vectors #rvv‑sl
The length of a vector \vec{a} is written either \|\vec{a}\| or just plain a. The length can be computed using Pythagorus’ theorem:
Pythagorus' length formula. #rvv‑ey
a = \|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2}
Derivation
Warning: Length must be computed in a single basis. #rvv‑wl
Computing the length of a vector using Pythagorus' theorem. #rvv‑fl
Some common integer vector lengths are \vec{a} = 4\hat\imath + 3\hat\jmath (length a = 5) and \vec{b} = 12\hat\imath + 5\hat\jmath (length b = 13).
Warning: Adding vectors does not add lengths. #rvv‑wa
Did you know?
Sets of three integers a,b,c where a^2 + b^2 = c^2 are called Pythagorean triples. A long list of such triples is given on the Plimpton 322 clay tablet written by the ancient Babylonians around 1800 BCE, although it is unclear how they generated these numbers. Pythagorean triples lead to complex mathematics, including the curious patterns shown below and Fermat's Last Theorem.
The values of a and b for all Pythagorean triples a,b,c with a and b up to 2000.
Unit vectors #rvv‑su
A unit vector is any vector with a length of one. We use the special over-hat notation \hat{a} to indicate when a vector is a unit vector. Any non-zero vector \vec{a} gives a unit vector \hat{a} that specifies the direction of \vec{a}.
Normalization to unit vector. #rvv‑eu
\begin{aligned} \hat{a} = \frac{\vec{a}}{a}\end{aligned}
Derivation
Any vector can be written as the product of its length and direction:
Vector decomposition into length and direction. #rvv‑ei
\begin{aligned} \vec{a} = a \hat{a}\end{aligned}
Derivation
Three vectors and their decompositions into lengths and directional unit vectors. #rvv‑fu
Vectors and units #rvv‑sn
When using vectors to describe physical quantities we need to have the correct units associated with them, just as for position coordinates. Basis vectors such as \hat\imath,\hat\jmath,\hat{k} have no units (they are also dimensionless), so the components must have units. For example, a velocity vector \vec{v} = (4\hat\imath + 3\hat\jmath){\rm\ m/s} has components v_1 = 4{\rm\ m/s} and v_2 = 3{\rm\ m/s}, which then multiply the dimensionless (and unit-less) basis vectors \hat\imath and \hat\jmath. For convenience, when all components of a vector have the same units, we will often write the units just once at the end, so that all of the following expressions are equivalent: \begin{aligned} \vec{v} &= (4{\rm\ m/s})\,\hat\imath + (3{\rm\ m/s})\,\hat\jmath \\ &= (4\hat\imath + 3\hat\jmath){\rm\ m/s} \\ &= 4\hat\imath + 3\hat\jmath {\rm\ m/s}. \end{aligned}
When a vector is expressed as \vec{a} = a \hat{a}, decomposed into a length and direction, the length a has units but the direction unit vector \hat{a} has no units. The terminology here is slightly confusing, as unit vectors have no units. For example, if we calculate the length of \vec{v} above we obtain: v = \sqrt{v_1^2 + v_2^2} = \sqrt{(4{\rm\ m/s})^2 + (3{\rm\ m/s})^2} = \sqrt{25{\rm\ m^2/s^2}} = 5{\rm\ m/s}. Then the unit vector is \hat{v} = \frac{\vec{v}}{v} = \frac{(4{\rm\ m/s})\,\hat\imath + (3{\rm\ m/s})\,\hat\jmath}{5{\rm\ m/s}} = 0.8\,\hat\imath + 0.6\,\hat\jmath.
Dot Product #rvv‑sd
The dot product (also called the inner product or scalar product) is defined by
Dot product from components. #rvv‑es
\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3
An alternative expression for the dot product can be given in terms of the lengths of the vectors and the angle between them:
Dot product from length/angle. #rvv‑ed
\vec{a} \cdot \vec{b} = a b \cos\theta
Derivation
The fact that we can write the dot product in terms of components as well as in terms of lengths and angle is very helpful for calculating the length and angles of vectors from the component representations.
Length and angle from dot product. #rvv‑el
\begin{aligned} a &= \sqrt{\vec{a} \cdot \vec{a}} \\ \cos\theta &= \frac{\vec{b} \cdot \vec{a}}{b a}\end{aligned}
Derivation
If two vectors have zero dot product \vec{a} \cdot \vec{b} = 0 then they have an angle of \theta = 90^\circ = \frac{\pi}{2}\rm\ rad between them and we say that the vectors are perpendicular, orthogonal, or normal to each other.
In 2D we can easily find a perpendicular vector by rotating \vec{a} counterclockwise with the following equation.
Counterclockwise perpendicular vector in 2D. #rvv‑en
\vec{a}^\perp = -a_2\,\hat\imath + a_1\hat\jmath
Derivation
In 2D there are two perpendicular directions to a given vector \vec{a}, given by \vec{a}^\perp and -\vec{a}^\perp. In 3D there is are many perpendicular vectors, and there is no simple formula like #rvv-en for 3D.
The perpendicular vector \vec{a}^\perp is always a +90^\circ rotation of \vec{a}. #rvv‑fn
Related applications
Cross Product #rvv‑sc
The cross product can be defined in terms of components by:
Cross product in components. #rvv‑ex
\vec{a} \times \vec{b} = (a_2 b_3 - a_3 b_2) \,\hat{\imath} + (a_3 b_1 - a_1 b_3) \,\hat{\jmath} + (a_1 b_2 - a_2 b_1) \,\hat{k}
It is sometimes more convenient to work with cross products of individual basis vectors, which are related as follows.
Cross products of basis vectors. #rvv‑eo
\begin{aligned} \hat\imath \times \hat\jmath &= \hat{k} & \hat\jmath \times \hat{k} &= \hat\imath & \hat{k} \times \hat\imath &= \hat\jmath \\ \hat\jmath \times \hat\imath &= -\hat{k} & \hat{k} \times \hat\jmath &= -\hat\imath & \hat\imath \times \hat{k} &= -\hat\jmath \\ \end{aligned}
Derivation
Warning: The cross product is not associative. #rvv‑wc
Rather than using components, the cross product can be defined by specifying the length and direction of the resulting vector. The direction of \vec{a} \times \vec{b} is orthogonal to both \vec{a} and \vec{b}, with the direction given by the right-hand rule. The magnitude of the cross product is given by:
Cross product length. #rvv‑el
\| \vec{a} \times \vec{b} \| = a b \sin\theta
Derivation
This second form of the cross product definition can also be related to the area of a parallelogram.
The area of a parallelogram is the length of the base multiplied by the perpendicular height, which is also the magnitude of the cross product of the side vectors. #rvv‑fx
A useful special case of the cross product occurs when vector \vec{a} is in the 2D \hat\imath,\hat\jmath plane and the other vector is in the orthogonal \hat{k} direction. In this case the cross product rotates \vec{a} by 90^\circ counterclockwise to give the perpendicular vector \vec{a}^\perp, as follows.
Cross product of out-of-plane vector \hat{k} with 2D vector \vec{a} = a_1\,\hat\imath + a_2\,\hat\jmath. #rvv‑e9
\hat{k} \times \vec{a} = \vec{a}^\perp
Derivation
Projection and complementary projection #rvv‑so
The projection and complementary projection are:
Projection of \vec{a} onto \vec{b}. #rvv‑ep
\operatorname{Proj}(\vec{a},\vec{b}) = (\vec{a} \cdot \hat{b}) \hat{b} = (a \cos\theta) \, \hat{b}
Complementary projection of \vec{a} with respect to \vec{b}. #rvv‑em
\begin{aligned} \operatorname{Comp}(\vec{a}, \vec{b}) &= \vec{a} - \operatorname{Proj}(\vec{a}, \vec{b}) = \vec{a} - (\vec{a} \cdot \hat{b}) \hat{b} \\ \left\| \operatorname{Comp}(\vec{a}, \vec{b}) \right\| &= a \sin\theta \end{aligned}
Adding the projection and the complementary projection of a vector just give the same vector again, as we can see on the figure below.
Projection of \vec{a} onto \vec{b} and the complementary projection. #rvv‑fm
As we see in the diagram above, the complementary projection is orthogonal to the reference vector:
Complementary projection is orthogonal to the reference. #rvv‑er
\operatorname{Comp}(\vec{a}, \vec{b}) \cdot \vec{b} = 0
Derivation
Changing bases #rvv‑sb
To change the basis that a vector is written in, we need to know how the basis vectors are related. We do this by writing one set of basis vectors in terms of the other basis vectors. If we want to change from \hat\imath,\hat\jmath to \hat{u},\hat{v}, then we need to write \hat\imath,\hat\jmath in terms of \hat{u},\hat{v} and then substitute the expressions.
Example: Basis change. #rvv‑xn
We can also write the general expressions for basis change, as below.
Change of basis formulas. #rvv‑eg
\begin{aligned} a_i &= a_u u_i + a_v v_i + a_w w_i & a_u &= a_i i_u + a_j j_u + a_k k_u \\ a_j &= a_u u_j + a_v v_j + a_w w_j & a_v &= a_i i_v + a_j j_v + a_k k_v \\ a_k &= a_u u_k + a_v v_k + a_w w_k & a_w &= a_i i_w + a_j j_w + a_k k_w \end{aligned}
Derivation
In 2D the change between two orthonormal bases is a rotation by an angle \theta, resulting in the change of basis expression below.
Change of basis formula in 2D. #rvv‑e2
\begin{aligned} a_i &= \cos\theta \, a_u - \sin\theta \, a_v & a_u &= \cos\theta \, a_i + \sin\theta \, a_j \\ a_j &= \sin\theta \, a_u + \cos\theta \, a_v & a_v &= -\sin\theta \, a_i + \cos\theta \, a_j \end{aligned}
Derivation
Vector expressions are true no matter which basis we write the vectors in, even if they are written in different bases.
Example: Vector addition in different bases. #rvv‑xa
Example Problem: Cross product in different bases. #rvv‑xx
Example: Dot product is independent of basis. #rvv‑xd
Time-dependent vectors and bases #rvv‑st
We can have dynamic vectors which change over time, so their components also change. Alternatively, we can have a fixed vector but dynamic basis. Because we are using orthonormal bases, the only type of basis change that can occur is rigid rotational motion of the basis.
Movement: vector basis
A time-varying vector or basis. Whether the vector or the basis changes with time, in either case the components are also changing with time. #rvv‑fi