Definition: $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$.
Geometric form: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$
where $\theta \in [0, \pi]$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.
Properties:
- Commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
- Distributive: $\mathbf{a} \cdot (\mathbf{b}+\mathbf{c}) = \mathbf{a}\cdot\mathbf{b} + \mathbf{a}\cdot\mathbf{c}$
- $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$
- $\mathbf{a} \perp \mathbf{b} \iff \mathbf{a} \cdot \mathbf{b} = 0$ (for non-zero $\mathbf{a}, \mathbf{b}$)
Angle between two vectors:
$$\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$$
Example 1: Find the angle between $\mathbf{a} = \mathbf{i}+2\mathbf{j}+2\mathbf{k}$ and $\mathbf{b} = 2\mathbf{i}-\mathbf{j}+2\mathbf{k}$.
$\mathbf{a}\cdot\mathbf{b} = 2 - 2 + 4 = 4$. $|\mathbf{a}| = 3$, $|\mathbf{b}| = 3$.
$$\cos\theta = \frac{4}{9} \Rightarrow \theta = \arccos\left(\frac{4}{9}\right) \approx 63.6^\circ$$
Scalar projection of $\mathbf{b}$ onto $\mathbf{a}$: $\text{comp}_{\mathbf{a}}\mathbf{b} = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|}$.
Vector projection of $\mathbf{b}$ onto $\mathbf{a}$: $\text{proj}_{\mathbf{a}}\mathbf{b} = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|^2}\mathbf{a}$.
Definition (3D only):
$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\ a_1 & a_2 & a_3\ b_1 & b_2 & b_3\end{vmatrix}
= (a_2 b_3 - a_3 b_2)\mathbf{i} - (a_1 b_3 - a_3 b_1)\mathbf{j} + (a_1 b_2 - a_2 b_1)\mathbf{k}$$
Geometric form: $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$.
$\mathbf{a} \times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (by the right-hand rule).
Properties:
- Anti-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$
- $\mathbf{a} \times \mathbf{a} = \mathbf{0}$
- $\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} \times \mathbf{b} = \mathbf{0}$ (for non-zero vectors)
- Distributive: $\mathbf{a} \times (\mathbf{b}+\mathbf{c}) = \mathbf{a}\times\mathbf{b} + \mathbf{a}\times\mathbf{c}$
Example 2: Compute $\mathbf{a} \times \mathbf{b}$ for $\mathbf{a} = \mathbf{i}+2\mathbf{j}$ and $\mathbf{b} = 3\mathbf{i}-\mathbf{j}+\mathbf{k}$.
$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\1&2&0\3&-1&1\end{vmatrix}
= (2\cdot1-0\cdot(-1))\mathbf{i} - (1\cdot1-0\cdot3)\mathbf{j} + (1\cdot(-1)-2\cdot3)\mathbf{k}
= 2\mathbf{i} - \mathbf{j} - 7\mathbf{k}$$
Area of parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$: $|\mathbf{a} \times \mathbf{b}|$.
Area of triangle with two sides $\mathbf{a}$ and $\mathbf{b}$: $\tfrac{1}{2}|\mathbf{a} \times \mathbf{b}|$.
$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix}a_1&a_2&a_3\b_1&b_2&b_3\c_1&c_2&c_3\end{vmatrix}$$
Geometric meaning: Volume of the parallelepiped with edges $\mathbf{a}, \mathbf{b}, \mathbf{c}$.
$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are coplanar iff $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0$.
KEY TAKEAWAY: The dot product produces a scalar and measures alignment; the cross product produces a perpendicular vector and measures the area of the parallelogram. Both are essential tools for geometry and mechanics in 3D.
EXAM TIP: Expand the cross product using the determinant form — expand along the top row. Double-check signs (the $\mathbf{j}$ term has a minus sign in front).
COMMON MISTAKE: Applying the cross product in 2D (it is only defined in 3D, or as a scalar in the 2D special case $a_1 b_2 - a_2 b_1$).
