Further Maths - Dot Product
Pearson Edexcel Further Mathematics 2022
Definition and computation
What is the word explanation for the scalar/dot product of two vectors?
The sum of the products of the components.
What’s the notation for the dot product of $\pmb{a}$ and $\pmb{b}$?
What’s the sum formula for $\pmb{a} \cdot \pmb{b}$?
\[ \left(\begin{matrix} 2 \\ 2 \\ 2 \end{matrix}\right) \cdot \left(\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}\right) \]
What is the dot product of the two vectors?
Perpendicularity and intuition
What does it mean if the dot product of two vectors is zero?
The two vectors are perpendicular.
\[\pmb{a} \cdot \pmb{b} = 0\]
What is true about $\pmb{a}$ and $\pmb{b}$?
They are perpendicular.
What’s the intuition behind the dot product?
The closer it is to zero, the more different the vectors are.
The cosine formula
What is the $\cos$ formula for the dot product of $\pmb{a}$ and $\pmb{b}$?
\[\pmb{a} \cdot \pmb{b} = \vert \pmb{a} \vert \times \vert \pmb{b} \vert \times \cos\theta\]
What does $\theta$ represent here?
The angle between two vectors $\pmb{a}$ and $\pmb{b}$.
\[\pmb{a} \cdot \pmb{b} = \vert \pmb{a} \vert \times \vert \pmb{b} \vert \times \cos\theta\]
What does $ \vert \pmb{a} \vert $ represent here?
The length of vector $\pmb{a}$
\[\pmb{a} \cdot \pmb{b} = \vert \pmb{a} \vert \times \vert \pmb{b} \vert \times \cos\theta\]
Can you make $\cos$ the subject of the formula?
\[\pmb{a} \cdot \pmb{b} = \vert \pmb{a} \vert \times \vert \pmb{b} \vert \times \cos\theta\]
Why must a value of $0$ mean the two vectors are perpendicular?
Because $\cos(90^{\circ}) = 0$.
\[\cos\theta = \frac{\pmb{a} \cdot \pmb{b}}{ \vert \pmb{a} \vert \vert \pmb{b} \vert }\]
What do the two inverses of $\cos$ mean?
- One inverse is the acute angle
- One inverse is the obtuse angle
What’s the formula for $\cos\theta$?
What’s the formula for $\cos\theta$?