Study sheet: Vectors and Vector Subspaces

📋 Course Outline

1. Vectors in the Plane and Space
2. Vector Operations and Linear Combinations
3. Coordinates, Norms, and Distances
4. Alignment, Bases, and Coplanarity
5. Dot Product and Orthogonal Projection
6. Determinants in the Plane
7. Cross Product in Space
8. Mixed Products and Volumes
9. Spanned Subspaces and Normal Vectors
10. Cartesian Equations of Vector Sets

📖 1. Vectors in the Plane and Space

🔑 Key Concepts & Definitions

• Real plane : the set of ordered pairs of real numbers
• Real space : the set of ordered triples of real numbers
• Geometric vector : associated with two distinct points A and B and has the magnitude of segment AB, the direction of line AB, and the orientation from A to B

💡 Memory Hook

Points are locations, whereas vectors encode directed displacement.

📖 2. Vector Operations and Linear Combinations

🔑 Key Concepts & Definitions

• Vector equality : if and only if they have the same magnitude, direction, and orientation
• Linear combination : A vector $\vec w$ is a linear combination of vectors $\vec u$ and $\vec v$ if there exist real numbers λ and μ such that $\vec w=\lambda\vec u+\mu\vec v$.

📝 Essential Points

• The sum of two nonzero vectors with different directions is represented by the diagonal of the parallelogram having those vectors as sides.

📌 For a real scalar λ and vector $\vec u$, positive λ preserves orientation and multiplies the magnitude by λ, negative λ reverses orientation, and $0\vec u=\vec 0$.

💡 Memory Hook

Place vectors head-to-tail or as parallelogram sides.

📖 3. Coordinates, Norms, and Distances

📝 Essential Points

• In an orthonormal frame, the standard basis of R2 is $(\vec i,\vec j)$, while the standard basis of R3 is $(\vec i,\vec j,\vec k)$.

📐 Formula — For a vector with coordinates $(x,y)$ in R2, its magnitude is $\|\vec u\|=\sqrt{x^2+y^2}$.

📐 Formula — For points A and B, the coordinate vector is $\overrightarrow{AB}=(x_B-x_A,y_B-y_A)$ in R2 and $\overrightarrow{AB}=(x_B-x_A,y_B-y_A,z_B-z_A)$ in R3.

💡 Memory Hook

Coordinates → vector difference → norm → distance.

📖 4. Alignment, Bases, and Coplanarity

🔑 Key Concepts & Definitions

• Vector basis : Two non-collinear vectors form a basis of R2, and three non-coplanar vectors form a basis of R3, because every vector has a unique linear combination of them.

📝 Essential Points

📌 Two vectors are collinear when one is a real scalar multiple of the other, whereas three vectors in R3 are coplanar when one is a linear combination of the other two.

💡 Memory Hook

Collinear vectors generate a line; non-coplanar vectors generate all of R3.

📖 5. Dot Product and Orthogonal Projection

🔑 Key Concepts & Definitions

• Orthogonality : if and only if their dot product is zero

📝 Essential Points

📐 Formula — For vectors $\vec u=(a_1,a_2)$ and $\vec v=(b_1,b_2)$ in R2, or corresponding three-coordinate vectors in R3, the dot product is $\vec u\cdot\vec v=a_1b_1+a_2b_2$ in R2 and $\vec u\cdot\vec v=a_1b_1+a_2b_2+a_3b_3$ in R3.

📐 Formula — The orthogonal projection of $\vec u$ onto a nonzero vector $\vec v$ is $p_{\vec v}(\vec u)=\frac{\vec u\cdot\vec v}{\|\vec v\|^2}\vec v$.

📐 Formula — For nonzero vectors with angle θ, the dot product satisfies $\vec u\cdot\vec v=\|\vec u\|\|\vec v\|\cos(\theta)$.

💡 Memory Hook

Dot product zero → orthogonality; projection splits a vector into parallel and perpendicular parts.

📖 6. Determinants in the Plane

📝 Essential Points

📐 Formula — For $\vec u=(a_1,a_2)$ and $\vec v=(b_1,b_2)$ in R2, the determinant is $\det(\vec u,\vec v)=a_1b_2-a_2b_1$.

📐 Formula — The area of the parallelogram spanned by two plane vectors is $A=|\det(\vec u,\vec v)|$.

💡 Memory Hook

Dot products measure angles; determinants measure oriented area.

📖 7. Cross Product in Space

🔑 Key Concepts & Definitions

• Cross product : the unique vector perpendicular to both inputs, with magnitude $\|\vec u\wedge\vec v\|=\|\vec u\|\|\vec v\|\sin(\theta)$ and orientation determined by the right-hand rule

📝 Essential Points

📐 Formula — For vectors with coordinates $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$, their cross product is $\vec u\wedge\vec v=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$.

📐 Formula — The norm of the cross product equals the area of the parallelogram spanned by the two vectors: $\|\vec u\wedge\vec v\|=\|\vec u\|\|\vec v\||\sin(\theta)|$.

💡 Memory Hook

Perpendicular direction → right-hand orientation → parallelogram area.

📖 8. Mixed Products and Volumes

🔑 Key Concepts & Definitions

• Mixed product : the scalar triple product $\det(\vec u,\vec v,\vec w)=\vec u\cdot(\vec v\wedge\vec w)$

📝 Essential Points

📐 Formula — The absolute value of the mixed product is the volume of the parallelepiped spanned by three vectors: $V=|\det(\vec u,\vec v,\vec w)|$.

📌 Three vectors in R3 form a basis if and only if their mixed-product determinant is nonzero.

💡 Memory Hook

A base parallelogram lifted by a height forms a parallelepiped.

📖 9. Spanned Subspaces and Normal Vectors

🔑 Key Concepts & Definitions

• Spanned subspace : the set of all linear combinations $\operatorname{Span}(\vec u_1,\ldots,\vec u_n)=\{\lambda_1\vec u_1+\cdots+\lambda_n\vec u_n\mid \lambda_i\in\mathbb R\}$
• Normal vector : orthogonal to every vector in that subspace

📝 Essential Points

📌 If one vector in a generating family is a linear combination of the others, removing it does not change the spanned subspace.

💡 Memory Hook

Linear combinations generate a subspace; orthogonality to generators gives a normal vector.

📖 10. Cartesian Equations of Vector Sets

🔑 Key Concepts & Definitions

• Vector line in R2 : the set $\operatorname{Span}(\vec u)$, a vector line in R2 with direction vector $\vec u$
• Vector plane in R3 : the set $\operatorname{Span}(\vec u_1,\vec u_2)$ generated by two non-collinear vectors

📝 Essential Points

📐 Formula — The vector line generated by $\vec u=(a,b)$ has Cartesian equation $ay-bx=0$.

📐 Formula — If a vector plane has Cartesian equation $ax+by+cz=0$, then $(a,b,c)$ is a normal vector to the plane.

💡 Memory Hook

A vector line is one-dimensional, a vector plane is two-dimensional, and both pass through the origin.

📊 Synthesis Tables

Plane and space vector criteria

Object	Criterion	Geometric meaning
Two vectors in R2	det(u,v) ≠ 0	They form a basis
Three vectors in R3	det(u,v,w) ≠ 0	They form a basis
Two vectors in R2	det(u,v) = 0	They are collinear
Three vectors in R3	det(u,v,w) = 0	They are coplanar

Geometric products

Product	Output	Main interpretation
Dot product	Scalar	Angle and orthogonality
Determinant in R2	Scalar	Oriented parallelogram area
Cross product in R3	Vector	Normal direction and area
Mixed product in R3	Scalar	Signed parallelepiped volume