2021-01-14
What is the word explanation for the scalar/dot produt of two vectors??
The sum of the products of the components.
What’s the notation for the dot product of $\pmb{a}$ and $\pmb{b}$??
$$ \pmb{a} \cdot \pmb{b} $$
What’s the sum formula for $\pmb{a} \cdot \pmb{b}$??
$$ \sum \pmb{a}_i \pmb{b}_i $$
$$ \left(\begin{matrix} 2 \ 2 \ 2 \end{matrix}\right) \cdot \left(\begin{matrix} 1 \ 2 \ 3 \end{matrix}\right) $$ What is the dot product of the two vectors??
$$ 12 $$
What is $\hat{i} \cdot \hat{i}$??
$$ 1 $$
What is $\hat{j} \cdot \hat{j}$??
$$ 1 $$
Does the dot product give a vector or scalar answer??
Scalar.
What does it mean if the dot product of two vectors is zero??
The two vectors are perpindicular.
$$\pmb{a} \cdot \pmb{b} = 0$$ What is true about $\pmb{a}$ and $\pmb{b}$??
They are perpindicular.
What’s the intuition behind the dot product??
The closer it is to zero, the more different the vectors are.
What is the $\cos$ formula for the dot product of $\pmb{a}$ and $\pmb{b}$??
$$ \pmb{a} \cdot \pmb{b} = |\pmb{a}| \times |\pmb{b}| \times \cos\theta $$
$$\pmb{a} \cdot \pmb{b} = |\pmb{a}| \times |\pmb{b}| \times \cos\theta$$ What does $\theta$ represent here??
The angle between two vectors $\pmb{a}$ and $\pmb{b}$.
$$\pmb{a} \cdot \pmb{b} = |\pmb{a}| \times |\pmb{b}| \times \cos\theta$$ What does $|\pmb{a}|$ represent here??
The length of vector $\pmb{a}$
$$\pmb{a} \cdot \pmb{b} = |\pmb{a}| \times |\pmb{b}| \times \cos\theta$$ Can you make $\cos$ the subject of the formula??
$$ \cos\theta = \frac{\pmb{a} \cdot \pmb{b}}{|\pmb{a}||\pmb{b}|} $$
$$\pmb{a} \cdot \pmb{b} = |\pmb{a}| \times |\pmb{b}| \times \cos\theta$$ Why must a value of $0$ mean the two vectors are perpindicular??
Because $\cos(90^{\circ}) = 0$.
$$\cos\theta = \frac{\pmb{a} \cdot \pmb{b}}{|\pmb{a}||\pmb{b}|}$$ What do the two inverses of $\cos$ mean??
- One inverse is the actute angle
- One inverse is the obtuse angle
What’s the formula for $\cos\theta$??
$$ \cos\theta = \frac{\pmb{a} \cdot \pmb{b}}{|\pmb{a}||\pmb{b}|} $$
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date: 2021-01-14 14:01
tags:
- '@?further-maths'
- '@?vectors'
- '@?public'
title: Further Maths - Dot Product
