I want to discuss basic properties of vectors here: How to add them, subtract them, multiply (there are many ways), divide (there is no standard way) and other things. Let's start with a list of legal operations and illegal ones.[br][br]LEGAL OPERATIONS & PROPERTIES[br][center][math][br]\vec{a}+\vec{b} = \vec{b}+\vec{a} \text{ : commutative law}\\[br]\vec{a}+(\vec{b}+\vec{c}) = (\vec{a}+\vec{b})+\vec{c} \text{ : associative law} \\[br]\vec{a}-\vec{b} = -(\vec{b}-\vec{a}) \\[br]d\vec{a} =\vec{a}d \text{ : scalar multiplication by a vector} \\[br]0\vec{a} = \vec{0} \text{ : definition of the zero vector}\\[br]\vec{a}+\vec{0} = \vec{a} \text{ : additive identity} \\[br]\vec{a}+(-\vec{a})=\vec{a}-\vec{a}=\vec{0} \text{ : additive inverse}\\[br]\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a} \text{ : the scalar or dot product which we'll discuss in a later chapter}\\[br]\vec{a}\times \vec{b} = -\vec{b}\times \vec{a} \text{ : the vector or cross product which we'll discuss in a later chapter}\\[br][/math][/center][br][br]ILLEGAL OPERATIONS[br][center][math][br]\vec{a}/\vec{b} \text{ : division of a vector by a vector is undefined}\\[br]d/\vec{b} \text{ : division of a scalar by a vector is undefined} \\[br]\vec{a}\;\vec{b} \text{ : conventional multiplication is undefined}\\[br]\vec{a}+d \text{ : illegal addition of a scalar and a vector} \\[br]\vec{a}+0 = \text{ : same rule} \\[br][/math][/center][br][br]

In the list of illegal operations of vectors, you see both conventional multiplication and division. That should cause concern. How could we use the mathematics of vectors to describe our world without basic concepts like multiplication and division? The short answer is we cannot. So that's certainly not the end of the story.[br][br]The longer answer is that there is no way to divide vectors. Some sources will suggest that it doesn't exist because there's no need for it. That's nonsense. Division is supposed to be an inverse of multiplication. So the explanation goes more like this: There is no singular way to define the product of vectors, and so the inversion of that process isn't possible. Another way of saying this is that while the inverse of a real numbers x is just 1/x, and while the inverse of a complex number z is always defined - although it takes a tiny bit of effort to find it (see multiplicative inverse of complex numbers), this operation cannot be done with vectors. It is known to be impossible. [br][br]Where does that leave us. If the inverse of a vector doesn't exist, then division doesn't exist, and multiplication becomes hazy. In practice there are two versions of multiplication - the dot product (also called the scalar product), and the cross product (also called the vector product). The dot product returns a scalar result and will be used in our definition of work in an upcoming chapter. The cross product will be used in a lot of rotational problems like those involving torque, angular momentum, Coriolis force, etc. There is no defined operation like [math]\vec{a}\vec{b}[/math] that we could imagine calculating with a FOIL-like method for the three components of vector a times the three of vector b. The result is a mess that can't be inverted. If you want to read more about this, look into the topic of normed division algebras. [br][br]So since the form of multiplication of vectors must be specified, we must place a symbol between the vectors to know whether a dot or cross product is intended. Those look like [math]\vec{a} \cdot \vec{b}[/math] or [math]\vec{a} \times \vec{b}.[/math]

[color=#a61c00]AN ASIDE: While it is possible to construct inverses for real and complex numbers, and while it isn't possible to construct the inverse of a 3D vector, it is possible to construct the inverse of higher dimensional numbers. What's interesting is that the only allowable dimensions are 4 (quaternions) and 8 (octonians). Outside of these, there are more complicated infinite-dimensional options, but no others. While this will have nothing at all to do with this course, it seems to be the case that our seemingly 3D world is not actually adequately described by 3D vectors. Proper descriptions of nature seem to require a higher dimensional space. The implications of this are the stuff of sci-fi novels and philosophy of religion papers! To put it another way: How can it be that our world runs on mathematics that suggests the existence of things that don't seem to exist in our world... everything from extra dimensions to imaginary numbers? Such is our predicament.[/color]

When adding or subtracting vectors, it must be done by components just as when adding complex numbers the real gets added to real and imaginary to imaginary. Here x gets added to x and y to y and z to z.[br][color=#1e84cc][br]EXAMPLE: Let's add [math]\vec{a}=2\hat{i}+3\hat{j}[/math] and [math]\vec{b}=-1\hat{i}+4\hat{j}.[/math][br]SOLUTION: [math]\vec{a}+\vec{b}=(2-1)\hat{i}+(3+4)\hat{j} = 1\hat{i}+7\hat{j}.[/math][br][br]EXAMPLE: Using the same vectors, let's find [math]\vec{b}-\vec{a}.[/math][br]SOLUTION: [math]\vec{b}-\vec{a}=(-1-2)\hat{i}+(4-3)\hat{j}=-3\hat{i}+1\hat{j}.[/math][br][br]EXAMPLE: Again with the same vectors, let's find [math]3\vec{a}-2\vec{b}.[/math][br]SOLUTION: [math]3(2\hat{i}+3\hat{j})-2(-1\hat{i}+4\hat{j})=6\hat{i}+9\hat{j}+2\hat{i}-8\hat{j}=8\hat{i}+1\hat{j}.[/math][br][/color]