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zlun%|Fg=XNnBpE7tG2yFN#`SBpTdwOgft?K8EA|QcgR|NotDNSa_9+q8~``T0gjUh zc0mR%7J`0Sjqhl0*Fd9B`i^15*wNP)J*(gfa9l;ejH-r>uxP$GG6a2(bMj=S1qMx1 zz5#M_c$f=not-kE5>cX})S!bLV>Adf(Xen8=vh)m#(0>MVIcxfmaqCB&wE1{ZZVab zmSzDlA~l~G;_~kwb(L2_1VBu#-rgImj#3naP8@}~kQr9OK4JJ44hTY28HjyG>O5@! z9=H7KVhZ~E9A+`K_4O8)e@`)1c@ag<@|oHazP_-i2+J}SSyBwCjI@;%8x&4rdU_-n z)YiE As pointed out, the projection and component actually refers to the same thing. +> To solve a problem like this it useful to introduce a coordinate system, as you mentioned yourself +> you project onto the x-axis. +> As soon as you introduce a coordinate system you can talk about the components of some vector. + +## Position Vectors + +If an origin $O$ is fixed, then any point $P$ in space may be represented by the vector $\pmb r$ +which has a magnitude and direction given by the line $\overrightarrow{OP}$. + +A point $(x, y, z)$ in Cartesian space has the position vector $r = x\pmb i + y\pmb j + z\pmb k$. + +# Applications of Vectors + +## Application of Vectors to Geometry + +### Equation of a Straight Line + +A straight line can be specified by + +- two points it passes +- one point it passes and a direction + +If $\pmb a$ and $\pmb b$ are the position vectors of two distinct points, then the position vectors +of an arbitrary point on the line joining these points is: + +$$\pmb r = \pmb a + \lambda(\pmb b - \pmb a)$$ + +where $\lambda \in \Re$ is a parameter. + +![](./images/vimscrot-2022-02-18T21:55:30,367159917+00:00.png) + +Suppose $O$ is an origin and $\pmb a$, $\pmb b$, and $\pmb r$ are position vectors on the line such +that + +\begin{align*} +\pmb a &= (x_0, y_0, z_0) \\ +\pmb b &= (x_1, y_1, z_1) \\ +\pmb r &= (x, y, z)\\ +\\ +(x, y, z) &= (x_0, y_0, z_0) + \lambda((x_1, y_1, z_1) - (x_0, y_0, z_0)) \\ +\\ +x &= x_0 + \lambda(x_1-x_0) \\ +y &= y_0 + \lambda(y_1-y_0) \\ +z &= z_0 + \lambda(z_1-z_0) \\ +\\ +\lambda &= \frac{x-x_0}{x_1-x_0} = \frac{y-y_0}{y_1-y_0} = \frac{z-z_0}{z_1-z_0} +\end{align*} + +In the above, the vector $\pmb b - \pmb a$ is in the direction of the line. +Thus the equation of a line can be specified by giving a point it passes through ($\pmb a$, say) and +the direction of the line ($\pmb d = (d_1, d_2, d_3)$, say). + +The vector equation is then + +$$\pmb r = \pmb a + \lambda\pmb d$$ + +#### The Cartesian Equation + +$$\frac{x-x_0}{d_1} = \frac{y-y_0}{d_2} = \frac{z-z_0}{d_3}$$ + +### Equation of a Plane + +A *plane* can be defined by specifying either: + +- three points (as long as they're not in a straight line) +- a point on the plne and two directions (useful for a parametric form) +- specifying a point on the plane and the normal vector to the plane + +#### Specifying a Point and a Normal Vector + +Let $\pmb a$ be the position vector of a point on the plane, and $\pmb n$ a normal vector to the +plane. + +If $\pmb r$ is the position vector of an arbitrary point on the plane, then $\pmb r - \pmb a$ is a +vector lying *in* the plane, so + +$$(\pmb r - \pmb a) \cdot \pmb n = 0$$ + +So the *vector equation* of the plane is + +$$\pmb r \cdot \pmb n = \pmb a \cdot n = d$$ + +where $\pmb r = (x, y, z)$ and the vectors $\pmb a$ and $\pmb n$ are known. + +Suppose $\pmb a$, $\pmb n$, and $\pmb r$ are given by + +\begin{align*} +\pmb a &= (x_0, y_0, z_0) \\ +\pmb n &= (l, m, p) \\ +\pmb n &= (x, y, z)\\ +\text{then } 0 &= ((x, y, z) - (x_0, y_0, z_0))\cdot(l, m, p) +\end{align*} + +#### Specifying Three Points on a Plane + +If we specify three points on a plane with position vectors $\pmb a$, $\pmb b$, and $\pmb c$ the +vectors $\pmb c - \pmb a$ and $\pmb c - \pmb b$ lie *in* the plane. +(The vectors $\pmb a$, $\pmb b$, and $\pmb c$ do not necessarily lie *in* the plane; +rather they take you from $O$ **to** the plane.) + +The normal to the plane, $\pmb n$, is then parallel to + +$$(\pmb c - \pmb a)\times(\pmb c - \pmb b)$$ + +and so the equation of the plane is + +$$(\pmb r - \pmb a)\cdot((\pmb c - \pmb a)\times(\pmb c - \pmb b)) = 0$$ + +#### The Angle Between Two Planes + +... is the same as the angle between their normal vectors + + + + + +[^d_associative]: The grouping of elements in in an operation do not matter (e.g. scalar addition: $a+(b+c) = (a+b)+c$) +[^d_commutative]: The order of elements in an operation do not matter (e.g. scalar addition: $a+b = b+a$) +[^d_distributive]: Easiest to explain with examples. Scalar multiplication is said to be distributive because $(a+b)c = ac + bc$