# Further Maths - Dot Product

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/vector-dot-product/ · Updated: 2021-01-14 · Tags: further-maths, vectors

### 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

##### ![PHOTO DOT PRODUCT ANGLE](dot-product-angle.png) What's the formula for $\cos\theta$??
$$
\cos\theta = \frac{\pmb{a} \cdot \pmb{b}}{|\pmb{a}||\pmb{b}|}
$$

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