Vector Operations Notes (2D Vectors, Dot and Cross Product)

Vector representation in 2D
- A vector a can be written as $\mathbf{a} = ax \hat{\mathbf{i}} + ay \hat{\mathbf{j}}.$
- Similarly, $\mathbf{b} = bx \hat{\mathbf{i}} + by \hat{\mathbf{j}}.$
- When adding vectors, add corresponding components:
- $\mathbf{c} = \mathbf{a} + \mathbf{b} \Rightarrow cx = ax + bx,\; cy = ay + by.$
- If a component is missing (zero), treat it as zero in the addition.
- Subtraction is component-wise: $\mathbf{a} - \mathbf{b} = (ax - bx)\hat{\mathbf{i}} + (ay - by)\hat{\mathbf{j}} = \mathbf{a} + (-\mathbf{b}).$
Dot product (2D and 3D)
- For 2D vectors $\mathbf{a} = ax \hat{\mathbf{i}} + ay \hat{\mathbf{j}}, \quad \mathbf{b} = bx \hat{\mathbf{i}} + by \hat{\mathbf{j}},$
 $\mathbf{a} \cdot \mathbf{b} = ax bx + ay by.$
- In 3D, with $\mathbf{a} = ax \hat{\mathbf{i}} + ay \hat{\mathbf{j}} + az \hat{\mathbf{k}}, \; \mathbf{b} = bx \hat{\mathbf{i}} + by \hat{\mathbf{j}} + bz \hat{\mathbf{k}},$
 the dot product is $\mathbf{a} \cdot \mathbf{b} = ax bx + ay by + az bz.$
- Basis dot products (important for intuition):
- $\hat{\mathbf{i}} \cdot \hat{\mathbf{i}} = 1, \quad \hat{\mathbf{j}} \cdot \hat{\mathbf{j}} = 1,\quad \hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = 0.$
- Geometric interpretation: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\cos\theta,$ where $\theta$ is the angle between the vectors.
- Therefore, $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}.$
- Projections: the component of $\mathbf{b}$ along $\mathbf{a}$ satisfies $|\mathbf{b}|\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|}.$
Magnitude of a vector
- $|\mathbf{a}| = \sqrt{ax^2 + ay^2 + a_z^2}.$
- In 2D, this becomes $|\mathbf{a}| = \sqrt{ax^2 + ay^2}.$
Cross product (3D) and its relation to 2D
- For 3D vectors, $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ ax & ay & az \ bx & by & bz \end{vmatrix}$
 which yields a vector perpendicular to both a and b.
- Key component relations: $\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}},\quad \hat{\mathbf{j}}\times\hat{\mathbf{i}} = -\hat{\mathbf{k}}.$
- In 2D problems, the cross product is often discussed via its z-component: $ax by - ay bx$ , which equals the magnitude of the 2D parallelogram area and points in the ±z direction.
- Orthogonality check: $\hat{\mathbf{i}} \times \hat{\mathbf{i}} = \mathbf{0},\quad \hat{\mathbf{j}} \times \hat{\mathbf{j}} = \mathbf{0}.$
- Note from transcript: cross product is “opposite” in behavior to the dot product: dot product uses parallel alignment (cos θ), cross product uses perpendicular orientation (sin θ) for magnitude.
Worked example 1: dot product and magnitudes (A = 3i + 2j, B = 0i + 8j)
- Define components: $\mathbf{a} = 3\hat{\mathbf{i}} + 2\hat{\mathbf{j}}, \quad \mathbf{b} = 0\hat{\mathbf{i}} + 8\hat{\mathbf{j}}.$
- Dot product: $\mathbf{a} \cdot \mathbf{b} = (3)(0) + (2)(8) = 16.$
- Magnitudes: $|\mathbf{a}| = \sqrt{3^2 + 2^2} = \sqrt{13} \approx 3.606, \quad |\mathbf{b}| = \sqrt{0^2 + 8^2} = 8.$
- Angle between: $\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|} = \frac{16}{(\sqrt{13})(8)} \approx 0.5547,$ hence $\theta = \arccos(0.5547) \approx 56.0^{\circ}.$
- Projection of b onto a: $|\mathbf{b}|\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|} = \frac{16}{\sqrt{13}} \approx 4.441.$
- Cross product (magnitude, 3D viewpoint): $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}|\,|\mathbf{b}|\sin\theta = (\sqrt{13})(8)\sin(56.0^{\circ}) \approx 24,$ and in 3D this would be the vector $\mathbf{a} \times \mathbf{b} = (0,0,24).$
- Common pitfall noted in transcript corrections: the speaker incorrectly stated some magnitudes (e.g., claiming |a| = 5). The correct magnitude is $\sqrt{13} \approx 3.606.$
Worked example 2: quick cross-check from a common 2D setup (A = 2i + 1j, B = 2i + 1j or similar)
- General method: add components to get the resultant, or apply dot/cross as needed.
- Takeaway: addition is component-wise, dot product uses component-wise multiplication, and cross product in 3D gives a vector perpendicular with magnitude |a||b|sinθ.
Key takeaways and practical implications
- Always treat vectors via their components; add/subtract component-wise: cx = ax ± bx, cy = ay ± by, etc.
- Dot product provides a measure of how aligned two vectors are: if θ = 0°, a · b = |a||b|; if θ = 90°, a · b = 0.
- The dot product is a scalar; the cross product is a vector (in 3D) perpendicular to the plane of the inputs.
- The magnitude and angle relationships are linked: a · b = |a||b| cosθ and |a × b| = |a||b| sinθ (in 3D, with cross product vector direction given by the right-hand rule).
- Projections are derived from the dot product; one common form is the component of b along a: $|\mathbf{b}|\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|}.$
Quick reference formulas (for review)
- Vector: $\mathbf{a} = ax \hat{\mathbf{i}} + ay \hat{\mathbf{j}} + a_z \hat{\mathbf{k}}.$
- Dot product (2D): $\mathbf{a} \cdot \mathbf{b} = ax bx + ay by.$
- Dot product (3D): $\mathbf{a} \cdot \mathbf{b} = ax bx + ay by + az bz.$
- Magnitude: $|\mathbf{a}| = \sqrt{ax^2 + ay^2 + a_z^2}.$
- Angle: $\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|},\quad \theta = \arccos\left(\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}\right).$
- Projection of b onto a: $|\mathbf{b}|\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|}.$
- Cross product (3D): $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ ax & ay & az \ bx & by & bz \end{vmatrix}.$
- 2D cross-product magnitude (z-component): $ax by - ay bx.$
Note on the transcript content
- The speaker mixed several concepts and sometimes produced incorrect numerical results (e.g., misidentifying magnitudes). The formulas and workflows above reflect standard vector algebra and correct values for the examples shown. If you’re ever unsure about a computed magnitude or angle, recalculate from the definitions above using the components.

Vector Representation
- A vector $\mathbf{a}$ can be written as components: $\mathbf{a} = ax \hat{\mathbf{i}} + ay \hat{\mathbf{j}} + a*z \hat{\mathbf{k}}.$
- Vector addition/subtraction is component-wise: $\mathbf{c} = \mathbf{a} \pm \mathbf{b} \Rightarrow cx = ax \pm bx,\; cy = ay \pm by$ .
Dot Product
- Scalar product for 2D vectors $\mathbf{a} \cdot \mathbf{b} = ax bx + ay by$ , and for 3D $\mathbf{a} \cdot \mathbf{b} = ax bx + ay by + az bz$ .
- Geometric interpretation: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}\,|\mathbf{b}|\cos\theta$ , where $heta$ is the angle between vectors. It measures alignment.
- Used to find the angle between vectors: $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}\,|\mathbf{b}|}$ .
- Projections: the component of $\mathbf{b}$ along $\mathbf{a}$ is $\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|}$ .
Magnitude of a Vector
- The length of a vector $\mathbf{a}$ is $\text{|}\mathbf{a}| = \sqrt{ax^2 + ay^2 + a_z^2}$ (for 3D) or $\text{|}\mathbf{a}| = \sqrt{ax^2 + ay^2}$ (for 2D).
Cross Product (primarily 3D)
- $\mathbf{a} \times \mathbf{b}$ yields a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (in 3D).
- Calculated using a determinant: $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ ax & ay & az \ bx & by & bz \end{vmatrix}$ .
- In 2D, the cross product's magnitude (z-component) is $ax by - ay bx$ , representing the area of the parallelogram formed by the vectors.
- The cross product's magnitude is related by $\text{|}\mathbf{a} \times \mathbf{b}| = |\mathbf{a}\,|\mathbf{b}|\sin\theta$ .
Key Principles
- Vector operations (addition, subtraction, dot product) are performed component-wise.
- The dot product results in a scalar, while the cross product (in 3D) results in a vector.
- Dot product indicates vector alignment (parallels), cross product indicates perpendicular orientation (magnitude related to sine).
Quick Reference Formulas
- Vector: $\mathbf{a} = ax \hat{\mathbf{i}} + ay \hat{\mathbf{j}} + a*z \hat{\mathbf{k}}$
- Dot product (3D): $\mathbf{a} \cdot \mathbf{b} = ax bx + ay by + az bz$
- Magnitude: $\text{|}\mathbf{a}| = \sqrt{ax^2 + ay^2 + a*z^2}$
- Angle: $\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}\,|\mathbf{b}|}$
- Cross product (3D): $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ ax & ay & az \ bx & by & bz \end{vmatrix}$