Study guide: Vector Algebra and Calculus

📋 Course Outline

1. Vector Definition and Classification
2. Components and Orthonormal Bases
3. Vector Addition and Scalar Multiplication
4. Linear Combinations and Independence
5. Scalar Product and Vector Norm
6. Orthogonality and Orthonormality
7. Vector Product
8. Mixed and Double Vector Products
9. Vector Projections and Division
10. Sine Rule for Triangles
11. Gradient, Divergence, and Curl

📖 1. Vector Definition and Classification

🔑 Key Concepts & Definitions

• Vector: A directed segment with an origin and endpoint, defined by its origin, direction, sense, and magnitude.
• Free vector: A specified direction, sense, and magnitude, but an unspecified supporting line and point of application.
• Sliding vector: A point of application that is not fixed.
• Bound vector: All its elements determined.
• Unit vector: A vector whose magnitude equals 1.

📝 Essential Points

• The vector space considered is the usual three-dimensional Euclidean space.

📖 2. Components and Orthonormal Bases

🔑 Key Concepts & Definitions

• Orthonormal basis: A basis is orthonormal when its basis vectors are mutually perpendicular and each has magnitude 1.
• Direct basis: A basis in which, viewed from the endpoint of e₃, the rotation from e₁ toward e₂ is counterclockwise.

★ Must-know

📌 In an orthonormal basis, the dot product of basis vectors eᵢ and eⱼ equals 1 when i = j and 0 when i ≠ j.

📐 Formula - In a three-dimensional basis, a vector with components (x, y, z) is $\vec{V}=x\vec{e}_1+y\vec{e}_2+z\vec{e}_3$.

Further detail

• The real numbers x, y, and z are the components of vector V in the stated basis.

📖 3. Vector Addition and Scalar Multiplication

🔑 Key Concepts & Definitions

• Vector sum: The sum of two vectors in R³ is a vector in R³ whose components are the sums of the corresponding components of the addends.

★ Must-know

• For vectors with components (a₁, a₂, a₃) and (b₁, b₂, b₃), their sum has components $(a_1+b_1,\ a_2+b_2,\ a_3+b_3)$.

• Multiplying a vector V by a real scalar λ produces a vector collinear with V, with each component multiplied by λ.

Further detail

📐 Formula - Vector addition is commutative: $\vec{V}_1+\vec{V}_2=\vec{V}_2+\vec{V}_1$.

📐 Formula - Vector addition is associative: $(\vec{V}_1+\vec{V}_2)+\vec{V}_3=\vec{V}_1+(\vec{V}_2+\vec{V}_3)$.

• The zero vector has components (0, 0, 0) and is the additive identity: adding it to a vector leaves that vector unchanged.

📐 Formula - Every vector V has an opposite −V such that $\vec{V}+(-\vec{V})=\vec{0}$.

📐 Formula - Scalar multiplication distributes over scalar addition: $(\lambda_1+\lambda_2)\vec{V}=\lambda_1\vec{V}+\lambda_2\vec{V}$.

📐 Formula - Scalar multiplication is associative: $\lambda_1(\lambda_2\vec{V})=(\lambda_1\lambda_2)\vec{V}$.

📖 4. Linear Combinations and Independence

🔑 Key Concepts & Definitions

• Linear combination: A linear combination of vectors V₁ through Vₙ is a vector of the form $\vec{W}=\sum_{i=1}^{n}\lambda_i\vec{V}_i$, where each λᵢ is real.
• Linear independence: Vectors are linearly independent if the relation $\sum_{i=1}^{n}\lambda_i\vec{V}_i=\vec{0}$ implies that every coefficient λᵢ is zero.
• Linear dependence: Vectors are linearly dependent when a linear combination equal to the zero vector has coefficients that are not all zero.

📝 Essential Points

• A single nonzero vector is linearly independent.

• No linearly independent system can contain the zero vector.

• Every subset of a linearly independent system is linearly independent.

• If a system of vectors is linearly dependent, at least one vector in the system is a linear combination of the others.

💡 Memory Hook

Independence forces every coefficient to zero; dependence permits a nonzero relation.

📖 5. Scalar Product and Vector Norm

🔑 Key Concepts & Definitions

• Dot product: The dot product of two vectors is a real scalar equal to the product of their magnitudes and the cosine of the angle between them.

★ Must-know

• For vectors with components (a₁, a₂, a₃) and (b₁, b₂, b₃) in an orthonormal basis, their dot product is $a_1b_1+a_2b_2+a_3b_3$.

Further detail

📐 Formula - The dot product is linear in each vector, including $(\lambda\vec{V})\cdot\vec{W}=\lambda(\vec{V}\cdot\vec{W})$.

📐 Formula - The dot product is symmetric: $\vec{V}\cdot\vec{W}=\vec{W}\cdot\vec{V}$.

📌 The dot product is zero when the vectors are orthogonal or when at least one of them is the zero vector.

📐 Formula - The dot product of a nonzero vector with itself is positive: $\vec{V}\cdot\vec{V}>0$ when $\vec{V}\ne\vec{0}$.

📖 6. Orthogonality and Orthonormality

🔑 Key Concepts & Definitions

• Vector norm: The positive square root of the vector's dot product with itself.
• Orthogonal vectors: Two vectors are orthogonal if and only if their dot product is zero.

★ Must-know

📐 Formula - The norm satisfies $\|\vec{V}\|=\sqrt{\vec{V}\cdot\vec{V}}$.

Further detail

📐 Formula - For a real scalar λ, the norm of a scaled vector satisfies $\|\lambda\vec{V}\|=|\lambda|\|\vec{V}\|$.

📐 Formula - The triangle inequality is stated as $\|\vec{V}_1\|-\|\vec{V}_2\|\leq\|\vec{V}_1+\vec{V}_2\|\leq\|\vec{V}_1\|+\|\vec{V}_2\|$.

📌 Three nonzero vectors that are pairwise orthogonal are linearly independent and form an orthogonal basis in R³.

💡 Memory Hook

Orthogonality gives perpendicular vectors; orthonormality also requires unit lengths.

📖 7. Vector Product

🔑 Key Concepts & Definitions

• Vector product: The vector product of two vectors is a vector perpendicular to both, with magnitude equal to the product of their magnitudes and the sine of the angle between them.

★ Must-know

📐 Formula - The vector product is antisymmetric: $\vec{V}_1\wedge\vec{V}_2=-\vec{V}_2\wedge\vec{V}_1$.

Further detail

📌 The vector product is zero if the two vectors are collinear or if either vector is zero.

• The magnitude of the vector product equals the area of the parallelogram formed by its two input vectors.

• The vector product is distributive over vector addition in either argument.

📐 Formula - A real scalar can be factored out of either argument of a vector product: $(\lambda\vec{V})\wedge\vec{W}=\lambda(\vec{V}\wedge\vec{W})$.

📐 Formula - In a direct orthonormal basis, the cyclic products are $\vec{e}_1\wedge\vec{e}_2=\vec{e}_3$, $\vec{e}_2\wedge\vec{e}_3=\vec{e}_1$, and $\vec{e}_3\wedge\vec{e}_1=\vec{e}_2$.

📐 Formula - Reversing each cyclic unit-vector product changes its sign: $\vec{e}_2\wedge\vec{e}_1=-\vec{e}_3$, $\vec{e}_3\wedge\vec{e}_2=-\vec{e}_1$, and $\vec{e}_1\wedge\vec{e}_3=-\vec{e}_2$.

• In a direct orthonormal basis, the vector product of components (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂) has components $(Y_1Z_2-Z_1Y_2,\ Z_1X_2-X_1Z_2,\ X_1Y_2-Y_1X_2)$.

💡 Memory Hook

Cycle forward: e₁ × e₂ = e₃, e₂ × e₃ = e₁, e₃ × e₁ = e₂.

📖 8. Mixed and Double Vector Products

🔑 Key Concepts & Definitions

• Mixed product: The mixed product of V₁, V₂, and V₃ in that order is the scalar $\vec{V}_1\cdot(\vec{V}_2\wedge\vec{V}_3)$.
• Double vector product: The double vector product of three vectors is the vector $\vec{V}_1\wedge(\vec{V}_2\wedge\vec{V}_3)$.

★ Must-know

📐 Formula - The double vector product satisfies $\vec{A}\wedge(\vec{B}\wedge\vec{C})=\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})$.

Further detail

• The absolute value of the mixed product equals the volume of the parallelepiped formed by the three vectors.

📌 The mixed product is zero if the three vectors are coplanar, if two are collinear, or if one vector is zero.

📐 Formula - The mixed product is invariant under cyclic permutation: $\vec{V}_1\cdot(\vec{V}_2\wedge\vec{V}_3)=\vec{V}_3\cdot(\vec{V}_1\wedge\vec{V}_2)=\vec{V}_2\cdot(\vec{V}_3\wedge\vec{V}_1)$.

• The notation (V₁, V₂, V₃) is used as shorthand for the mixed product $\vec{V}_1\cdot(\vec{V}_2\wedge\vec{V}_3)$.

📖 9. Vector Projections and Division

🔑 Key Concepts & Definitions

• Projection onto an axis: The orthogonal projection of V onto an axis with unit vector u is the component $\vec{V}_u=(\vec{V}\cdot\vec{u})\vec{u}$.
• Projection onto a plane: For a plane with unit normal n, the orthogonal projection of V into the plane is $\vec{V}_\pi=\vec{V}-(\vec{V}\cdot\vec{n})\vec{n}$.

★ Must-know

📌 For the vector equation $\vec{X}\wedge\vec{V}=\vec{W}$ to have a solution, V must be nonzero and W must be orthogonal to V.

📐 Formula - When V is nonzero and W is orthogonal to V, all solutions of $\vec{X}\wedge\vec{V}=\vec{W}$ are $\vec{X}=\frac{\vec{V}\wedge\vec{W}}{\|\vec{V}\|^2}+\lambda\vec{V}$, where λ is real.

Further detail

📐 Formula - For a unit normal n, projection onto a plane can also be written $\vec{V}_\pi=\vec{n}\wedge(\vec{V}\wedge\vec{n})$.

📖 10. Sine Rule for Triangles

📝 Essential Points

📐 Formula - For triangle ABC with opposite angles α, β, and θ, the sine rule is $\frac{BC}{\sin\alpha}=\frac{AB}{\sin\beta}=\frac{AC}{\sin\theta}$.

📖 11. Gradient, Divergence, and Curl

🔑 Key Concepts & Definitions

• Gradient: The vector of its partial derivatives in the three coordinate directions.
• Divergence: The divergence of a vector field V is the scalar product $\operatorname{div}\vec{V}=\vec{\nabla}\cdot\vec{V}$.
• Curl: The curl of a vector field V is the vector product of the nabla operator and V: $\operatorname{rot}\vec{V}=\vec{\nabla}\wedge\vec{V}$.
• Nabla operator: In an orthonormal Cartesian frame, the nabla operator is $\vec{\nabla}=\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{k}$.

★ Must-know

📐 Formula - In an orthonormal Cartesian frame, $\operatorname{grad}U=\vec{\nabla}U=\frac{\partial U}{\partial x}\vec{i}+\frac{\partial U}{\partial y}\vec{j}+\frac{\partial U}{\partial z}\vec{k}$.

📐 Formula - For $\vec{V}=V_x\vec{i}+V_y\vec{j}+V_z\vec{k}$, $\operatorname{div}\vec{V}=\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}$.

Further detail

• For $U=3xy-2zx+5yz$, the gradient is $(3y-2z)\vec{i}+(3x+5z)\vec{j}+(-2x+5y)\vec{k}$.

📐 Formula - For a scalar field f and vector field A, $\operatorname{div}(f\vec{A})=f\operatorname{div}\vec{A}+\vec{A}\cdot\operatorname{grad}f$.

📐 Formula - For a scalar field f and vector field A, $\operatorname{rot}(f\vec{A})=\operatorname{grad}f\wedge\vec{A}+f\operatorname{rot}(\vec{A})$.

📐 Formula - The curl of a gradient is zero: $\operatorname{rot}(\operatorname{grad}f)=\vec{0}$.

📐 Formula - The divergence of a curl is zero: $\operatorname{div}(\operatorname{rot}\vec{A})=0$.

📐 Formula - $\operatorname{rot}(\operatorname{rot}\vec{A})=\operatorname{grad}(\operatorname{div}\vec{A})-\Delta\vec{A}$, where $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$.

📐 Formula - For vector fields A and B, $\operatorname{div}(\vec{A}\wedge\vec{B})=\vec{B}\cdot\operatorname{rot}(\vec{A})-\vec{A}\cdot\operatorname{rot}(\vec{B})$.

💡 Memory Hook

Gradient maps a scalar to a vector; divergence maps a vector to a scalar; curl maps a vector to a vector.