The dot product (also called the inner product or scalar product) is defined by
An alternative expression for the dot product can be given in terms of the lengths of the vectors and the angle between them:
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.
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.
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 \( \vec{a} \).
The cross product can be defined in terms of components by:
It is sometimes more convenient to work with cross products of individual basis vectors, which are related as follows.
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:
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.
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.
Time-dependent vectors can be differentiated in exactly the same way that we differentiate scalar functions. For a time-dependent vector \( \vec{a}(t) \), the derivative \( \dot{\vec{a}}(t) \) is:
Note that vector derivatives are a purely geometric concept. They don't rely on any basis or coordinates, but are just defined in terms of the physical actions of adding and scaling vectors.
Increment:
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Time:
Vector derivatives shown as functions of \(t\) and \(\Delta t\). We can hold \(t\) fixed and vary \(\Delta t\) to see how the approximate derivative \( \Delta\vec{a}/\Delta t \) approaches \( \dot{\vec{a}} \). Alternatively, we can hold \(\Delta t\) fixed and vary \(t\) to see how the approximation changes depending on how \( \vec{a} \) is changing.
We will use either the dot notation \( \dot{\vec{a}}(t) \) (also known as Newton notation) or the full derivative notation \( \frac{d\vec{a}(t)}{dt} \) (also known as Leibniz notation), depending on which is clearer and more convenient. We will often not write the time dependency explicitly, so we might write just \( \dot{\vec{a}} \) or \( \frac{d\vec{a}}{dt} \). See below for more details.
Most people know who Isaac Newton is, but perhaps fewer have heard of Gottfried Leibniz. Leibniz was a prolific mathematician and a contemporary of Newton. Both of them claimed to have invented calculus independently of each other, and this became the source of a bitter rivalry between the two of them. Each of them had different notation for derivatives, and both notations are commonly used today.
Leibniz notation is meant to be reminiscent of the definition of a derivative:
Newton notation is meant to be compact:
Note that a superscribed dot always denotes differentiation with respect to time \(t\). A superscribed dot is never used to denote differentiation with respect to any other variable, such as \(x\).
But what about primes? A prime is used to denote differentiation with respect to a function's argument. For example, suppose we have a function \(f=f(x)\). Then
Suppose we have another function \(g=g(s)\). Then
As you can see, while a superscribed dot always denotes differentiation with respect to time \(t\), a prime can denote differentiation with respect to any variable; but that variable is always the function's argument.
Sometimes, for convenience, we drop the argument altogether. So, if we know that \(y=y(x)\), then \(y'\) is understood to be the same as \(y'(x)\). This is sloppy, but it is very common in practice.
Each notation has advantages and disadvantages. The main advantage of Newton notation is that it is compact: it does not take a lot of effort to write a dot or a prime over a variable. However, the price you pay for convenience is clarity. The main advantage of Leibniz notation is that it is absolutely clear exactly which variable you are differentiating with respect to.
Notice how, with Leibniz notation, you can imagine the \(dx\)'s "cancelling out" on the right-hand side, leaving you with \(dy/dt\).
When thinking about vector derivatives, it is important to remember that vectors don't have positions. Even if a vector is drawn moving about, this is irrelevant for the derivative. Only changes to length and direction are important.
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Movement: bounce stretch circle twist slider rotate vertical fly
Vector derivatives for moving vectors. Vector movement is irrelevant when computing vector derivatives.
In a fixed basis we differentiate a vector by differentiating each component:
Time: | \(t = \) 0 s |
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Basis: | \( \hat\imath,\hat\jmath \) \( \hat{u},\hat{v} \) |
The vector derivative decomposed into components. This demonstrates graphically that each component of a vector in a particular basis is simply a scalar function, and the corresponding derivative component is the regular scalar derivative.
Leibniz notation is very convenient for remembering the chain rule. Consider the following examples of the chain rule in the two notations:
Notice how, with Leibniz notation, you can imagine the \(dx\)'s "cancelling out" on the right-hand side, leaving you with \(dy/dt\).
The chain rule also applies to vector functions. This is helpful for parameterizing vectors in terms of arc-length \(s\) or other quantities different than time \(t\).
Cartesian:
Polar:
Basic idea:
The Riemann-sum definition of the vector integral is:
In the above definition \( \vec{S}_N \) is the sum with \(N\) intervals, written here using the left-hand edge \( \tau_i \) in each interval.
Time:
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Integral of a vector function \( \vec{a}(t) \), together with the approximation using a Riemann sum.
Just like vector derivatives, vector integrals only use the geometric concepts of scaling and addition, and do not rely on using a basis. If we do write a vector function in terms of a fixed basis, then we can integrate each component:
Steps: