Dot Product | Notion

Table of Contents

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Definition

The dot product is a form of vector multiplication that results in a scalar value. dot product is sometimes calling as the “scalar product”.

Here is two vectors, $v = (v_x, v_y, v_z)$ and $u = (u_x, u_y, u_z)$.

Dot product is defined as follows.

$$ v · u = v_xu_x + v_yu_y + v_zu_z $$

Dot product has a geometric meaning realted to the angle between two vectors.

$$ u⋅v=∣∣u∣∣∣∣v∣∣cosθ $$

Wait!

||u||, ||v|| = the magnitudes (Vector Length ) of $u$, $v$
$θ$ = the angle between $u$, $v$. $(0 ≤ θ ≤ π)$ </aside>
Using dot product, find the angle between two vectors.

$$ cosθ =(\frac{u⋅v}{∣∣u∣∣∣∣v∣∣}) $$
Inverse $cos$ into $arccos$.

$$ θ = arccos(\frac{u⋅v}{∣∣u∣∣∣∣v∣∣}) $$

Example

Dot Product(HBpencil) | Desmos

Here is $u=(3,4)$, $v=(7,-3)$.

Dot product
- $u⋅v = (3)(7) + (4)(-3) = 21 - 12 = 9$
Length(magnitude)
- $∣∣u∣∣ = \sqrt{3^2 + 4^2} = 5$, $∣∣v∣∣ = \sqrt{7^2 + (-3)^2} = \sqrt{58}$
Apply the formula.

$$ θ = arccos(\frac{9}{5⋅\sqrt{58}}) = 76.328..° $$

Dot product can be interpreted geometrically as the projection of one vector onto the other vector.

Projection of $a$ onto $b$

The projection of vector $a$ onto vector $b$ is shown as the horizontal dashed line along $b$.
The projection shows how much of vector $a$ extends along the same direction as vector $b$.

Length of Projection

This is the component of $||a||$ that points in the direction of $||b||$.

$$ ∣∣a∣∣cosθ $$
The dot product scales this projection by the magnitude of vector.

$$ a⋅b=∣∣a∣∣∣∣b∣∣cosθ $$
In other expression.

$$ ∣∣a∣∣cosθ = \frac{a⋅b}{∣∣b∣∣} $$

Now, you can get the length of the projection!

Vector of Projection

As have seen earlier, can get the length of the projection. Now, let’s calculate the position vector of the projection.

Length of projection

$$ \frac{a⋅b}{∣∣b∣∣} $$
Need a vector with pure direction. That's Unit Vectors

$$ \hat{b}=\frac{b}{||b||} $$
Combine length & unit vector of projection.

$$ projection(a) = \frac{a⋅b}{||b||^2}b $$

Example

Dot Product(HBpencil) | Desmos

Here is $u=(3,4)$, $v=(7,-3)$.

Dot product

$$ u⋅v = (3)(7) + (4)(-3) = 21 - 12 = 9 $$
Length(magnitudes)
- $||u|| = \sqrt{3^2 + 4^2} = 5$, $||v|| = \sqrt{7^2 + (-3)^2} = \sqrt{58}$
Apply projection of $u$ onto $v$

$$ projection(u) = \frac{u⋅v}{||v||^2}v = \frac{9}{58}(7,-3) $$

$$ = (1.086.., -0.465..) $$

Orthogonalization is a process to make the group of vectors perpendicular(⊥) to each other. and when normalized, into an orthonormal set.

Orthogonalization & Orthonormalization

Formula