From: Andrew Lorimer Date: Thu, 31 Jan 2019 02:04:00 +0000 (+1100) Subject: dot products and vector angles X-Git-Tag: yr12~270 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/a97a120e6075b89a786ad827545aa7735cf867e9 dot products and vector angles --- diff --git a/spec/vectors.md b/spec/vectors.md index 0a95e9e..0b6ea2a 100644 --- 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 +