Vectors in R2 and R3

• The content covers vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$, including vector spaces, scalar products, vector products, and equations for lines and planes.

The Vector Space $\mathbb{R}^n$

• $\mathbb{R}^n$ is the set of all n-tuples of real numbers $(x_1, x_2, \dots, x_n)$ where each $x_i$ is a real number.

Operations in $\mathbb{R}^n$

• Vector addition is defined as $u + v = (u_1 + v_1, u_2 + v_2, \dots, u_n + v_n)$ for $u, v \in \mathbb{R}^n$.
• Scalar multiplication is defined as $\alpha u = (\alpha u_1, \alpha u_2, \dots, \alpha u_n)$ for $u \in \mathbb{R}^n$ and $\alpha \in \mathbb{R}$.

Properties of Vector Addition and Scalar Multiplication

• Commutativity: $u + v = v + u$.
• Associativity: $(u + v) + w = u + (v + w)$.
• Existence of a zero vector: There exists $0 \in \mathbb{R}^n$ such that $u + 0 = u$.
• Existence of an additive inverse: There exists $-u \in \mathbb{R}^n$ such that $u + (-u) = 0$.
• Distributivity of scalar multiplication over vector addition: $\alpha(u + v) = \alpha u + \alpha v$.
• Distributivity of scalar multiplication over scalar addition: $(\alpha + \beta)u = \alpha u + \beta u$.
• Associativity of scalar multiplication: $(\alpha \beta)u = \alpha(\beta u)$.
• Identity element for scalar multiplication: $1u = u$.

Scalar Product (Dot Product)

• The scalar product of $u = (u_1, u_2, \dots, u_n)$ and $v = (v_1, v_2, \dots, v_n) \in \mathbb{R}^n$ is defined as $u \cdot v = u_1v_1 + u_2v_2 + \dots + u_nv_n = \sum_{i=1}^n u_i v_i$.

Properties of the Scalar Product

• Commutativity: $u \cdot v = v \cdot u.
• Distributivity: $u \cdot (v + w) = u \cdot v + u \cdot w.
• Compatibility with scalar multiplication: $(\alpha u) \cdot v = \alpha(u \cdot v) = u \cdot (\alpha v)$.
• Non-negativity: $u \cdot u \ge 0$ and $u \cdot u = 0$ if and only if $u = 0$.

Vector Norm

• The norm of a vector $u$ is defined as $|u| = \sqrt{u \cdot u} = \sqrt{u_1^2 + u_2^2 + \dots + u_n^2}$.

Distance Between Two Vectors

• The distance between $u$ and $v$ is defined as $d(u, v) = |u - v| = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 + \dots + (u_n - v_n)^2}$.

Angle Between Two Vectors

• The cosine of the angle $\theta$ between $u$ and $v$ is given by $\cos \theta = \frac{u \cdot v}{|u| |v|}$.

Orthogonality

• Vectors $u$ and $v$ are orthogonal if $u \cdot v = 0$, denoted as $u \perp v$.

Orthogonal Projection

• The orthogonal projection of $u$ onto $v$ is defined as $\text{proy}_v u = \frac{u \cdot v}{|v|^2} v$.

Orthogonal Component

• The orthogonal component of $u$ along $v$ is defined as $\text{comp}_v u = u - \text{proy}_v u = u - \frac{u \cdot v}{|v|^2} v$.

Pythagorean Theorem

• If $u$ and $v$ are orthogonal, then $|u + v|^2 = |u|^2 + |v|^2$.

Vector Product (Cross Product)

• For $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3) \in \mathbb{R}^3$, the cross product is $u \times v = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)$.
• This can also be represented as the determinant of a matrix with $i, j, k$ in the first row, components of $u$ in the second row, and components of $v$ in the third row.

Properties of the Vector Product

• Anti-commutativity: $u \times v = -(v \times u)$.
• Distributivity: $u \times (v + w) = u \times v + u \times w$.
• Compatibility with scalar multiplication: $(\alpha u) \times v = \alpha(u \times v) = u \times (\alpha v)$.
• $u \times 0 = 0 \times u = 0$.
• $u \times u = 0$.
• $u \cdot (u \times v) = 0$.
• $v \cdot (u \times v) = 0$.

Geometric Interpretation of Vector Product

• $|u \times v|$ represents the area of the parallelogram determined by $u$ and $v$.
• $|u \cdot (v \times w)|$ represents the volume of the parallelepiped determined by $u, v,$ and $w$.

Equations of Lines and Planes

Lines in $\mathbb{R}^2$

• Vector equation: $r(t) = P_0 + tv$, where $t \in \mathbb{R}$.
• Parametric equation: $x = x_0 + at$, $y = y_0 + bt$.
• Symmetric equation: $\frac{x - x_0}{a} = \frac{y - y_0}{b}$.
• General equation: $Ax + By + C = 0$.

Lines in $\mathbb{R}^3$

• Vector equation: $r(t) = P_0 + tv$, where $t \in \mathbb{R}$.
• Parametric equation: $x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$.
• Symmetric equation: $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.

Planes in $\mathbb{R}^3$

• Vector equation: $r(s, t) = P_0 + su + tv$, where $s, t \in \mathbb{R}$.
• Parametric equation: $x = x_0 + as + dt$, $y = y_0 + bs + et$, $z = z_0 + cs + ft$.
• General equation: $Ax + By + Cz + D = 0$.