Study guide: Vectors, Products, and Subspaces

📋 Course Outline

1. Vectors in the Plane and Space
2. Vector Operations and Linear Combinations
3. Coordinates, Norms, and Distances
4. Alignment, Coplanarity, and Bases
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. Vector Lines and Planes

📖 1. Vectors in the Plane and Space

🔑 Key Concepts & Definitions

• Real plane R2: the set of all ordered pairs of real numbers, written R2 = {(x,y) | x,y ∈ R}
• Real space R3: the set of all ordered triples of real numbers, written R3 = {(x,y,z) | x,y,z ∈ R}
• Geometric vector: characterized by its magnitude, direction, and orientation from the initial point A toward the terminal point B
• Vector equality: Two geometric vectors are equal if and only if they have the same magnitude, direction, and orientation

💡 Memory Hook

Points are locations; vectors encode displacement.

📖 2. Vector Operations and Linear Combinations

🔑 Key Concepts & Definitions

• Zero vector: the representative of all geometric vectors with magnitude zero, including every vector AA
• Linear combination: A vector w is a linear combination of vectors u and v if there exist real numbers λ and μ such that w = λu + μv.

📝 Essential Points

📌 For a real scalar λ, λu has magnitude |λ|∥u∥, keeps the orientation of u when λ > 0, reverses it when λ < 0, and equals the zero vector when λ = 0.

• For any points A, B, and C, Chasles’ relation is $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$.

💡 Memory Hook

Add, scale, then combine linearly.

📖 3. Coordinates, Norms, and Distances

🔑 Key Concepts & Definitions

• Orthonormal frame: An orthonormal frame in the plane is R = (O,i,j), and in space it is R = (O,i,j,k), where the standard vectors are unit and mutually orthogonal.

📝 Essential Points

📐 Formula — For points A(xA,yA) and B(xB,yB) in the plane, $\overrightarrow{AB}=(x_B-x_A,y_B-y_A)$; in space, $\overrightarrow{AB}=(x_B-x_A,y_B-y_A,z_B-z_A)$.

📐 Formula — The norm of a vector (x,y) in R2 is $\|(x,y)\|=\sqrt{x^2+y^2}$, while the norm of (x,y,z) in R3 is $\|(x,y,z)\|=\sqrt{x^2+y^2+z^2}$.

📐 Formula — The distance between two points equals the norm of their displacement vector: in R2, $AB=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}$, and in R3, $AB=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}$.

💡 Memory Hook

A vector travels from A to B by subtracting coordinates.

📖 4. Alignment, Coplanarity, and Bases

🔑 Key Concepts & Definitions

• Collinear vectors: one is a real scalar multiple of the other
• Coplanar vectors: one of them is a linear combination of the other two
• Basis: Two non-collinear vectors form a basis of R2, while three non-coplanar vectors form a basis of R3, and every vector has a unique linear combination in that basis.

📝 Essential Points

📌 Three points A, B, and C are aligned if and only if the vectors AB and AC are collinear.

💡 Memory Hook

Collinear vectors determine lines; non-coplanar vectors determine space.

📖 5. Dot Product and Orthogonal Projection

🔑 Key Concepts & Definitions

• Dot product: For u = (a1,a2) and v = (b1,b2) in R2, their dot product is $u\cdot v=a_1b_1+a_2b_2$; in R3 it is $u\cdot v=a_1b_1+a_2b_2+a_3b_3$.
• Orthogonality: if and only if their dot product is zero

📝 Essential Points

📐 Formula — For every vector u, $u\cdot u=\|u\|^2$.

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

💡 Memory Hook

A zero dot product causes orthogonality; projection isolates the parallel component.

📖 6. Determinants in the Plane

📝 Essential Points

📐 Formula — For u = (a1,a2) and v = (b1,b2), the determinant is $\det(u,v)=a_1b_2-a_2b_1$.

• The absolute value |det(u,v)| is the area of the parallelogram spanned by u and v.

📌 Two vectors in R2 are collinear if and only if their determinant is zero.

📌 Two vectors form a basis of R2 if and only if their determinant is nonzero.

💡 Memory Hook

The determinant measures the signed area of a parallelogram.

📖 7. Cross Product in Space

🔑 Key Concepts & Definitions

• Cross product: For non-collinear vectors u and v in R3, the cross product u ∧ v is the unique vector orthogonal to both, with norm $\|u\wedge v\|=\|u\|\|v\|\sin(\theta)$ and positive orientation determined by the right-hand rule.

📝 Essential Points

📐 Formula — For u = (a1,a2,a3) and v = (b1,b2,b3), $u\wedge v=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$.

• The norm of u ∧ v is the area of the parallelogram spanned by u and v in R3.

💡 Memory Hook

Perpendicular direction, area magnitude, right-hand orientation.

📖 8. Mixed Products and Volumes

🔑 Key Concepts & Definitions

• Mixed product: The mixed product of u, v, and w in R3 is the scalar triple product $\det(u,v,w)=u\cdot(v\wedge w)$.

📝 Essential Points

📐 Formula — For coordinate columns u = (a1,a2,a3), v = (b1,b2,b3), and w = (c1,c2,c3), $\det(u,v,w)=a_1(b_2c_3-b_3c_2)-a_2(b_1c_3-b_3c_1)+a_3(b_1c_2-b_2c_1)$.

• The absolute value of det(u,v,w) is the volume of the parallelepiped spanned by u, v, and w.

💡 Memory Hook

A triple product measures the signed volume of a parallelepiped.

📖 9. Spanned Subspaces and Normal Vectors

🔑 Key Concepts & Definitions

• Spanned subspace: The subspace spanned by vectors u1,...,un is the set of all linear combinations λ1u1 + ··· + λnun with real coefficients λ1,...,λn.
• Linear independence: none of its vectors is a linear combination of the others

📝 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

Spanning describes what vectors generate; normality describes what vectors are perpendicular.

📖 10. Vector Lines and Planes

📝 Essential Points

📐 Formula — The vector line spanned by u = (a,b) in R2 has Cartesian equation $ay-bx=0$.

📐 Formula — The vector plane spanned by two non-collinear vectors in R3 has an equation $ax+by+cz=0$, where (a,b,c) is a normal vector to the plane.

📐 Formula — A vector v belongs to the plane Span(u1,u2) if and only if $\det(u_1,u_2,v)=0$.

📐 Formula — A vector v = (x,y,z) belongs to the line Span(u) with u = (a,b,c) if and only if $v\wedge u=0$, equivalently $bz-cy=0,\ cx-az=0,\ ay-bx=0$.

• Every vector line in R3 is the intersection of two non-parallel vector planes.

💡 Memory Hook

Line in R3 = intersection of two non-parallel vector planes.

📊 Synthesis Tables

Geometric objects and equations

Object	Generation	Cartesian condition
Vector line in R2	Span(u)	ay − bx = 0
Vector plane in R3	Span(u1,u2)	ax + by + cz = 0
Vector line in R3	Span(u)	Two independent homogeneous plane equations