dot products and vector angles

diff --git a/spec/vectors.md b/spec/vectors.md
index 0a95e9eff152342b310bb2ed179f5cabfaf30907..0b6ea2a591579df2297f0fbb2caefea27642e9e6 100644 (file)
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -97,11 +97,16 @@ Produces a real number, not a vector.
 
 $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
 
+**on CAS:** `dotP([a b c], [d e f])`
+
 ## Scalar product properties
 
 1. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
 2. $\boldsymbol{a \cdot 0}=0$
 3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
+4. $\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1$
+5. If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular
+6. $\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2$
 
 For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:  
 $$\boldsymbol{a \cdot b}=\begin{cases}
@@ -121,8 +126,12 @@ If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsy
 
 ## Finding angle between vectors
 
+**positive direction**
+
 $$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$
 
+**on CAS:** `angle([a b c], [a b c])` (Action -> Vector -> Angle)
+
 
 ## Vector projections
 
@@ -133,7 +142,7 @@ $$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\b
 ## Vector proofs
 
 **Concurrent lines -** $\ge$ 3 lines intersect at a single point  
-**Collinear points -** $\ge$ 3 points lie on the same line
+**Collinear points -** $\ge$ 3 points lie on the same line ($\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 \lt \mu \lt 1$)
 
 Useful vector properties:
 
@@ -164,3 +173,5 @@ Direction of a vector can be given by the angles it makes with $\vec{i}, \vec{j}
 For $\vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
 $$\cos \alpha = {a_1 \over |\vec{a}|}, \quad \cos \beta = {a_2 \over |\vec{a}|}, \quad \cos \gamma = {a_3 \over |\vec{a}|}$$
 
+**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\vec{i} + b\vec{j} + c\vec{k}$ and $x$-axis
+