PLAIN AND CIVIL: 4.1. INTRODUCTION TO DOT PRODUCT

A simple way to differentiate dot product from cross product is that it results to only a number while cross product will always end up in a vector with both magnitude and direction.

ALGEBRAIC INTERPRETATION

Consider vectors a and b. They have the following Cartesian representation:

$\overrightarrow{a}=a_{i}+a_{j}+a_{k}$

$\overrightarrow{b}=b_{i}+b_{j}+b_{k}$

With these spatial dimensions, dot product of the vectors is defined as the summation of the corresponding components multiplied together.

$\overrightarrow{a}\bullet \overrightarrow{b}=a_{i}b_{i}+a_{j}b_{j}+a_{k}b_{k}$

PROPERTIES OF DOT PRODUCT

1. It is commutative: $\overrightarrow{a} \bullet \overrightarrow{b}=\overrightarrow{b} \bullet\overrightarrow{a}$

2. It is distributive: $\overrightarrow{a} \bullet \left (\overrightarrow{b} +\overrightarrow{c}\right )=\overrightarrow{a} \bullet\overrightarrow{b} +\overrightarrow{a} \bullet\overrightarrow{c}$

3. Scalars can be moved between vectors: $\left ( \lambda a \right )\bullet b=a\bullet\left ( \lambda b \right )=\lambda \left ( a\bullet b \right )$

4. The dot product of a vector with itself is the square of the length of the vector:

$a\bullet a=a^{2}$

GEOMETRIC INTERPRETATION

Dot product can also be defined geometrically with the introduction of the angle between the vectors:

$a\bullet b=\left | a \right |\left | b \right |cos\Theta$

It is quite obvious that from this equation, we can solve for the angle between two vectors using Dot Product.

Also, by focusing on the angle, the value of $cos\Theta$ is maximum when theta is $0^{o}$ and $180^{o}$ :

at $0^{o}$ : the dot product will be positive

at $90^{o}$ : the dot product will be zero

at $180^{o}$ : the dot product will be negative

In summary, we can say that the Dot Product measures how parallel vectors travel together.

PROJECTION

Dot product is also of help if we desire to measure how similar vectors travel together. We can measure the projection of one vector onto the other. Projection can simply be imagined as the shadow of a vector if you project light on one vector to another. It is the component of one vector along the axis of the other.

Projection of a vector is done by taking the dot product of the Cartesian representation of the vector and the unit vector of the axis: