What is dot product and cross product?

Dot product: For vectors a and b, a · b = |a||b|cosθ. Result is a scalar. Used for projections or checking orthogonality. Component form: a_x b_x + a_y b_y + a_z b_z. Cross product: a × b produces a vector perpendicular to both a and b. Magnitude = |a||b|sinθ. Used for torque, angular momentum, or surface normals in 3D.

Intuitively, dot product measures how aligned two vectors are; cross product gives a perpendicular vector whose magnitude equals the parallelogram area formed by the vectors. Dot → scalar, cross → vector. Use dot for work calculation, cross for torque or normals.

Think about the output. That's the most straightforward way to differentiate them. | Feature | Dot Product ($\mathbf{A} \cdot \mathbf{B}$) | Cross Product ($\mathbf{A} \times \mathbf{B}$) | | :--- | :--- | :--- | | Output Type | A Scalar (a number) | A Vector (with magnitude and direction) | | Geometric Meaning | Projection/Similarity (Are they aligned?) | Area/Perpendicularity (How much do they create a plane?) | | Dimensions | Can be used in any dimension (2D, 3D, 4D...) | Only defined for 3D vectors! | In programming (like game development), the dot product is used to check if an object is in front of or behind you (line of sight), and the cross product is used to calculate surface normals and rotation axes. They are two different lenses through which you analyze vector interactions!

Fantastic question! These are two of the most fundamental operations in Vector Algebra, used everywhere from physics to 3D graphics. ### 1. Dot Product (Scalar Product) * What it is: A way to multiply two vectors ($\mathbf{A}$ and $\mathbf{B}$) that results in a single scalar number (a magnitude, not a vector). * Formula: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta)$, where $\theta$ is the angle between the vectors. * What it tells you: It measures how much of one vector goes in the direction of the other. * If the result is positive, the vectors are generally pointing in the same direction ($\theta < 90^\circ$). * If the result is zero, the vectors are perpendicular ($\theta = 90^\circ$). * Real-world use: Calculating Work in physics (Work = Force $\cdot$ Distance). ### 2. Cross Product (Vector Product) * What it is: A way to multiply two vectors ($\mathbf{A}$ and $\mathbf{B}$) that results in a new vector ($\mathbf{C}$). * Direction: The resulting vector ($\mathbf{C}$) is perpendicular to both of the original vectors ($\mathbf{A}$ and $\mathbf{B}$). Its direction is determined by the Right-Hand Rule. * Magnitude (Length): The magnitude of the resulting vector ($|\mathbf{C}|$) is equal to the area of the parallelogram spanned by $\mathbf{A}$ and $\mathbf{B}$. * Formula: $\mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin(\theta) \mathbf{n}$, where $\mathbf{n}$ is the unit vector perpendicular to the plane of $\mathbf{A}$ and $\mathbf{B}$. * Real-world use: Calculating Torque in physics (Torque = Force $\times$ Lever Arm).