Upon completing this lesson, you will be able to:
The dot product is a way of multiplying two vectors that results in a scalar (a single number). It's useful for finding the angle between two vectors or for finding the component of one vector that acts along the direction of another.
The dot product, or scalar product, between two vectors $\vec{A}$ and $\vec{B}$ can be calculated in two ways:
Geometric Definition:
$$ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) $$
where $\theta$ is the angle between the two vectors. This is useful for finding the angle between two known vectors.
Component Definition:
$$ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z $$
This is the most common way to calculate the dot product.
A primary application in mechanics is finding the projection of a force onto an axis. The magnitude of the projection of a vector$\vec{A}$ onto a line defined by a unit vector $\mathbf{u}$ is simply $\vec{A} \cdot \mathbf{u}$. This tells you how much of the vector $\vec{A}$ is acting in the direction of $\mathbf{u}$.
The cross product is a way of multiplying two vectors that results in a new vector which is perpendicular to both of the original vectors. Its main application in statics is to calculate the moment of a force.
The cross product, or vector product, between two vectors $\vec{A}$ and $\vec{B}$ is written as:
$$ \vec{C} = \vec{A} \times \vec{B} $$
The resulting vector $\vec{C}$ has two key properties:
Direction:
The direction of $\vec{C}$ is perpendicular to the plane containing $\vec{A}$ and $\vec{B}$. This direction is determined by the right-hand rule: point the fingers of your right hand in the direction of the first vector $(\vec{A})$, then curl them towards the second vector $(\vec{B})$. Your thumb will point in the direction of $\vec{C}$. Because of this, the cross product is not commutative:
$$ \vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})
$$
Magnitude:
The magnitude of the cross product is given by $|\vec{C}| = |\vec{A}| |\vec{B}| \sin(\theta)$, where $\theta$ is the angle between the vectors.
For vectors given in Cartesian components:
$$ \vec{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k} \ \ \ \vec{B} = B_x\mathbf{i} + B_y\mathbf{j} + B_z\mathbf{k} $$
the cross product is most easily calculated using the determinant of a 3x3 matrix:
$$ \vec{A} \times \vec{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} $$
This expands to:
$$ \vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} $$
A moment is the turning effect a force has about a specific point. It is calculated by multiplying the force by the perpendicular distance from the point to the line of action of the force.
The moment of a force $\vec{F}$ about a point $O$, denoted $\vec{M}_O$, quantifies the tendency of the force to cause rotation around that point.
Scalar Method (2D):
The magnitude of the moment is
$$ M_O = F \cdot d $$
where $d$ is the moment arm—the shortest (perpendicular) distance from point $O$ to the line of action of the force.
The direction is assigned as positive $+$ for counter-clockwise (CCW) rotation and negative $-$ for clockwise (CW) rotation.
Vector Method (3D Cross Product):
The most formal method is to use the cross product:
$$ \vec{M}_O = \vec{r} \times \vec{F} $$
Here, $\vec{r}$ is a position vector drawn from the point $O$ to any point on the line of action of the force $\vec{F}$. This method automatically calculates both the magnitude and direction of the moment vector.