notes on vectors: corrections

uni/mmme/1026_maths_for_engineering/vectors.md

@@ -96,18 +96,19 @@ $i\cdot j = i\cdot k = j\cdot k = 0$.
 
 The vector product between two vectors is defined by:
 
-$$\pmb a \times \pmb b = |\pmb a||\pmb b|\sin\theta \pmb n$$
+$$\pmb a \times \pmb b = |\pmb a||\pmb b|\sin\theta \hat{\pmb n}$$
 
 where $0 \le \theta \le \pi$ is the angle between $\pmb a$ and $\pmb b$ and $\pmb n$ is a unit
 vector such that the three vectors from a right handed system:
 
 ![](./images/vimscrot-2022-02-18T20:11:12,072203286+00:00.png)
 
-- $\pmb a \times \pmb b = -\pmb b \times \pmb a$ (the vector product is non-commutative[^d_commutative])
+- $\pmb a \times \pmb b = -\pmb b \times \pmb a$ (the vector product is anti-commutative[^d_commutative])
 - If $\pmb a \times \pmb b = 0$ then either
 
-    i. The vectors are parralel
+    i. The vectors are parallel
     ii. One or both of the vectors are a zero vector
+
 - $(k_1\pmb a)\times(k_2\pmb b) = (k_1k_2)(\pmb a \times \pmb b)$ where $k_1$, $k_2$ are scalars
 - If $\pmb a = (a_1, a_2, a_3)$ and $\pmb b = (b_1, b_2, b_3)$ then
 
@@ -151,13 +152,13 @@ by those vectors:
 
 ## The Unit Vector
 
-$$\hat a = \frac{a}{|a|}$$
+$$\hat{\pmb a}= \frac{\pmb a}{|\pmb a|}$$
 
 ## Components of a Vector
 
-The component of a vector $\pmb a$ in the direction of the unit vector $\pmb n$ is
+The component of a vector $\pmb a$ in the direction of the **unit vector** $\hat{\pmb n}$ is
 
-$$\pmb a \cdot \pmb n$$
+$$\pmb a \cdot \hat{\pmb n}$$
 
 ![](./images/vimscrot-2022-02-18T20:34:50,128465689+00:00.png)
 
@@ -173,9 +174,9 @@ If $\pmb a = a_1\pmb i + a_2\pmb j + a_3\pmb k$ then the scalars $a_1$, $a_2$, a
 
 ### Vector Projections
 
-The *vector projection* of $\pmb a$ onto $\pmb n$ is given by
+The *vector projection* of $\pmb a$ onto $\hat{\pmb n}$ is given by
 
-$$(\pmb a \cdot \pmb n)\pmb n$$
+$$(\pmb a \cdot \hat{\pmb n})\hat{\pmb n}$$
 
 ![](./images/vimscrot-2022-02-18T21:40:15,724449945+00:00.png)
 
@@ -206,6 +207,10 @@ $$\cos\theta = \frac{\pmb a \cdot \pmb b}{|\pmb a||\pmb b|} = \frac{a_1b_1 + a_2
 
 ## Application of Vectors to Geometry
 
+### Area of a Parallelogram
+
+$$area = |\pmb a||\pmb b|\sin\theta = |\pmb a \times \pmb b|$$
+
 ### Equation of a Straight Line
 
 A straight line can be specified by
@@ -271,7 +276,7 @@ $$(\pmb r - \pmb a) \cdot \pmb n = 0$$
 
 So the *vector equation* of the plane is
 
-$$\pmb r \cdot \pmb n = \pmb a \cdot \pmb n = \pmb d$$
+$$\pmb r \cdot \pmb n = \pmb a \cdot \pmb n = D$$
 
 where $\pmb r = (x, y, z)$ and the vectors $\pmb a$ and $\pmb n$ are known.
 
@@ -280,7 +285,7 @@ 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)\\
+\pmb r &= (x, y, z)\\
 \text{then } 0 &= ((x, y, z) - (x_0, y_0, z_0))\cdot(l, m, p)
 \end{align*}