1.3 Dot Product and Cross Product

Dot products and cross products are the two fundamental ways to multiply vectors. The dot product outputs a scalar that tells you how aligned two vectors are, while the cross product outputs a new vector perpendicular to both inputs. Together, they let you calculate angles, areas, projections, work, and torque.

Vector Operations

Concept of Dot Product

The dot product has two equivalent forms. The algebraic form multiplies corresponding components and sums the results:

$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$

The geometric form connects the dot product to the angle between the vectors:

$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$

The fact that these two expressions are always equal is what makes the dot product so useful. The geometric form tells you what the dot product means: it measures how much one vector points in the direction of another. More precisely, $\mathbf{a} \cdot \mathbf{b}$ equals the length of $\mathbf{a}$'s projection onto $\mathbf{b}$, scaled by $|\mathbf{b}|$.

Three key properties:

• Commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
• Distributive: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$
• Scalar multiplication: $(k\mathbf{a}) \cdot \mathbf{b} = k(\mathbf{a} \cdot \mathbf{b})$

Note that the dot product is not associative. Writing $(\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c}$ doesn't even make sense, because $\mathbf{a} \cdot \mathbf{b}$ is a scalar, not a vector.

Calculation of Dot Product

To find the angle between two vectors, combine both forms of the dot product:

1. Compute $\mathbf{a} \cdot \mathbf{b}$ algebraically using components.
2. Compute $|\mathbf{a}|$ and $|\mathbf{b}|$.
3. Solve for the angle: $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$, then $\theta = \cos^{-1}\left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right)$.

Example: Find the angle between $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, -5, 6)$.

• $\mathbf{a} \cdot \mathbf{b} = (1)(4) + (2)(-5) + (3)(6) = 4 - 10 + 18 = 12$
• $|\mathbf{a}| = \sqrt{1 + 4 + 9} = \sqrt{14}$, $|\mathbf{b}| = \sqrt{16 + 25 + 36} = \sqrt{77}$
• $\cos\theta = \frac{12}{\sqrt{14}\sqrt{77}} = \frac{12}{\sqrt{1078}} \approx 0.365$, so $\theta \approx 68.6°$

Special cases to recognize quickly:

• If $\mathbf{a} \cdot \mathbf{b} = 0$, the vectors are perpendicular ($\theta = 90°$, since $\cos 90° = 0$).
• If $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|$, the vectors are parallel and same direction ($\theta = 0°$).
• If $\mathbf{a} \cdot \mathbf{b} = -|\mathbf{a}||\mathbf{b}|$, the vectors are parallel and opposite direction ($\theta = 180°$).
• A positive dot product means the angle is acute; a negative dot product means obtuse.

Cross Product

Definition of Cross Product

Unlike the dot product, the cross product produces a vector, not a scalar. The algebraic formula is:

$\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2,\; a_3b_1 - a_1b_3,\; a_1b_2 - a_2b_1)$

The magnitude of the result is given by the geometric form:

$|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$

Geometrically, the cross product gives you a vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. Its magnitude equals the area of the parallelogram formed by the two vectors. Where the dot product uses $\cos\theta$ to measure alignment, the cross product uses $\sin\theta$ to measure how "spread apart" the vectors are.

Three key properties:

• Anticommutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$ (order matters!)
• Distributive: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$
• Scalar multiplication: $(k\mathbf{a}) \times \mathbf{b} = k(\mathbf{a} \times \mathbf{b})$

The cross product is not commutative. Swapping the order flips the direction of the result. It's also not associative: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$ in general.

Computation of Cross Product

The component formula can be hard to memorize on its own. The determinant method gives you a systematic way to compute it:

$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Expand along the first row using cofactors:

1. The $\mathbf{i}$ component: $+(a_2b_3 - a_3b_2)$

2. The $\mathbf{j}$ component: $-(a_1b_3 - a_3b_1)$ (note the minus sign)

3. The $\mathbf{k}$ component: $+(a_1b_2 - a_2b_1)$

A common mistake is forgetting the negative sign on the $\mathbf{j}$ component. The signs alternate: $+, -, +$.

Direction is determined by the right-hand rule: point your fingers along $\mathbf{a}$, curl them toward $\mathbf{b}$, and your thumb points in the direction of $\mathbf{a} \times \mathbf{b}$.

Special case: If $\mathbf{a}$ and $\mathbf{b}$ are parallel (or one is the zero vector), then $\mathbf{a} \times \mathbf{b} = \mathbf{0}$, since $\sin 0° = \sin 180° = 0$.

Applications of Vector Products

Dot product applications:

• Work: $W = \mathbf{F} \cdot \mathbf{d}$, where $\mathbf{F}$ is force and $\mathbf{d}$ is displacement. This picks out only the component of force in the direction of motion.
• Scalar projection of $\mathbf{a}$ onto $\mathbf{b}$: $\text{comp}_{\mathbf{b}}\,\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$
• Orthogonality test: Two vectors are perpendicular if and only if their dot product is zero.

Cross product applications:

• Torque: $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$, where $\mathbf{r}$ is the position vector from the pivot to the point where force is applied.
• Area of a parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$: $A = |\mathbf{a} \times \mathbf{b}|$. The area of the triangle formed by those same vectors is $\frac{1}{2}|\mathbf{a} \times \mathbf{b}|$.
• Normal vectors: The cross product of two vectors in a plane gives a normal vector to that plane.

Two identities worth knowing:

• Scalar triple product: $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})$. This gives the (signed) volume of the parallelepiped formed by the three vectors.
• Vector triple product (BAC-CAB rule): $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$. The mnemonic "BAC-CAB" matches the order of vectors in the result.