Lesson 6: Vectors in Space

Lesson Objectives

Terminology:
- Vectors in space
Knowledge and Skills:
- Recognize vectors in space.
- Recognize and perform vector operations in space.

Vectors in Space

Basic Definitions and Concepts

Definition of a vector: A vector in space is a directed line segment.
Definition of magnitude: The magnitude (or length) of a vector in space is the distance between its initial point and its terminal point.
Line of action: The straight line passing through the initial point and terminal point of a vector is called the line of action (or supporting line) of that vector.
Collinear vectors: Two vectors are called collinear (or parallel) if their lines of action are parallel or coincident.
Direction of collinear vectors: If two vectors are collinear, they are either in the same direction or in opposite directions.
Equal vectors: Two vectors $\vec{a}$ and $\vec{b}$ are called equal, denoted by $\vec{a} = \vec{b}$ , if they have the same magnitude and the same direction.
Zero vector: Vectors whose initial point and terminal point coincide, such as $\overrightarrow{AA}, \overrightarrow{BB}, \dots$ , are called zero vectors.

Notes and Conventions

Point notation: A vector with initial point $A$ and terminal point $B$ is denoted by $\overrightarrow{AB}$ .
General notation: When it is not necessary to specify the initial point and terminal point of a vector, the vector is also denoted by $\vec{a}, \vec{b}, \vec{x}, \vec{y}, \dots$
Magnitude notation: The magnitude of vector $\overrightarrow{AB}$ is denoted by $|\overrightarrow{AB}|$ , and the magnitude of vector $\vec{a}$ is denoted by $|\vec{a}|$ .
Point determination property: In space, for any given point $O$ and vector $\vec{a}$ , there exists a unique point $M$ such that $\overrightarrow{OM} = \vec{a}$ .
Conventions on the zero vector: By convention, the zero vector has a magnitude of $0$ , and has the same direction (and therefore is collinear) with every vector. Consequently, all zero vectors are equal and denoted collectively by $\vec{0}$ .

Sum and Difference of Two Vectors in Space

a) Sum of Two Vectors in Space (Vector Addition)

Definition of the sum: In space, let $\vec{a}$ and $\vec{b}$ be two vectors. Choose an arbitrary point $A$ and points $B, C$ such that $\overrightarrow{AB} = \vec{a}$ and $\overrightarrow{BC} = \vec{b}$ . Then, the vector $\overrightarrow{AC}$ is called the sum of the two vectors $\vec{a}$ and $\vec{b}$ , denoted by $\vec{a} + \vec{b}$ .
Definition of vector addition: In space, the operation of finding the sum of two vectors is called vector addition.
Three-point rule (Triangle rule): If $A, B, C$ are three arbitrary points, then $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$ .
Parallelogram rule: If $ABCD$ is a parallelogram, then $\overrightarrow{AB} + \overrightarrow{AD} = \overrightarrow{AC}$ .
Parallelepiped rule (Box rule): For a parallelepiped $ABCD.A'B'C'D'$ , we have: $\overrightarrow{AB} + \overrightarrow{AD} + \overrightarrow{AA'} = \overrightarrow{AC'}$

Properties of Vector Addition

Commutative property: If $\vec{a}$ and $\vec{b}$ are two arbitrary vectors, then $\vec{a} + \vec{b} = \vec{b} + \vec{a}$ .
Associative property: If $\vec{a}, \vec{b}$ and $\vec{c}$ are three arbitrary vectors, then $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$ .
Addition with the zero vector: If $\vec{a}$ is any vector, then $\vec{a} + \vec{0} = \vec{0} + \vec{a} = \vec{a}$ .
Sum of multiple vectors: From the associative property of vector addition in space, we can write the sum of three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ as $\vec{a} + \vec{b} + \vec{c}$ without using parentheses. The same applies to the sum of multiple vectors in space.

b) Difference of Two Vectors in Space (Vector Subtraction)

Definition of opposite vector: In space, a vector having the same magnitude and opposite direction to vector $\vec{a}$ is called the opposite vector of vector $\vec{a}$ , denoted by $-\vec{a}$ .
Property of opposite vectors: Two vectors are opposite if and only if their sum equals $\vec{0}$ .
Notes on opposite vectors:
- The vector $\overrightarrow{BA}$ is an opposite vector of vector $\overrightarrow{AB}$ .
- The vector $\vec{0}$ is considered the opposite vector of itself.
Definition of the difference: The vector $\vec{a} + (-\vec{b})$ is called the difference of two vectors $\vec{a}$ and $\vec{b}$ , and is denoted by $\vec{a} - \vec{b}$ .
Definition of vector subtraction: In space, the operation of finding the difference of two vectors is called vector subtraction.
Subtraction rule: For any three points $O, A, B$ in space, we have: $\overrightarrow{OB} - \overrightarrow{OA} = \overrightarrow{AB}$

Multiplication of a Vector by a Scalar in Space

Definition

Definition of scalar multiplication: In space, the product of a real number $k \ne 0$ $k \neq = 0$ and a vector $\vec{a} \ne \vec{0}$ $a \neq = 0$ is a vector, denoted by $k\vec{a}$ $k a$ , defined as follows:
- In the same direction as vector $\vec{a}$ if $k > 0$ ; in the opposite direction to vector $\vec{a}$ if $k < 0$ ;
- Having magnitude equal to $|k| \cdot |\vec{a}|$ .
Definition of the operation: In space, the operation of finding the product of a number and a vector is called the multiplication of a vector by a scalar.

Notes and Condition for Collinearity

Conventions: $0\vec{a} = \vec{0}$ and $k\vec{0} = \vec{0}$ .
Zero product condition: If $k\vec{a} = \vec{0}$ , then $k = 0$ or $\vec{a} = \vec{0}$ .
Condition for collinearity: In space, a necessary and sufficient condition for two vectors $\vec{a}$ and $\vec{b}$ ( $\vec{b} \ne \vec{0}$ ) to be collinear is that there exists a real number $k$ such that $\vec{a} = k\vec{b}$ .

Properties of Multiplication by a Scalar

Associative property: If $h, k$ are two real numbers and $\vec{a}$ is any vector, then $h(k\vec{a}) = (hk)\vec{a}$ .
Distributive property with respect to scalar addition: If $h, k$ are two real numbers and $\vec{a}$ is any vector, then $(h + k)\vec{a} = h\vec{a} + k\vec{a}$ .
Distributive property with respect to vector addition: If $k$ is a real number and $\vec{a}, \vec{b}$ are two arbitrary vectors, then $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$ .
Multiplication by 1 and -1: If $\vec{a}$ is any vector, then $1\vec{a} = \vec{a}$ and $(-1)\vec{a} = -\vec{a}$ .

Dot Product of Two Vectors in Space

a) Angle Between Two Vectors in Space

Definition: In space, given two vectors $\vec{a}, \vec{b}$ distinct from $\vec{0}$ . Choose an arbitrary point $O$ and let $A, B$ be two points such that $\overrightarrow{OA} = \vec{a}, \overrightarrow{OB} = \vec{b}$ . Then, the angle $\widehat{AOB}$ ( $0^\circ \le \widehat{AOB} \le 180^\circ$ ) is called the angle between the two vectors $\vec{a}$ and $\vec{b}$ , denoted by $(\vec{a}, \vec{b})$ .
Perpendicular vectors: If the angle between two vectors $\vec{a}$ and $\vec{b}$ is $90^\circ$ , we say that the two vectors $\vec{a}$ and $\vec{b}$ are perpendicular (or orthogonal) to each other, denoted by $\vec{a} \perp \vec{b}$ .
Method to determine the angle: To determine the angle between two vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ in space, we can choose a point $E$ such that $\overrightarrow{AE} = \overrightarrow{CD}$ , then $(\overrightarrow{AB}, \overrightarrow{CD}) = \widehat{BAE}$ .
Convention on the angle: By convention, the angle between an arbitrary vector and $\vec{0}$ can take any value from $0^\circ$ to $180^\circ$ .

b) Dot Product of Two Vectors in Space

Definition and formula: In space, given two vectors $\vec{a}, \vec{b}$ both distinct from $\vec{0}$ . The dot product (or scalar product) of two vectors $\vec{a}$ and $\vec{b}$ is a real number, denoted by $\vec{a} \cdot \vec{b}$ , defined by the formula: $\vec{a} \cdot \vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot \cos(\vec{a}, \vec{b})$
Convention: By convention, if $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$ , then $\vec{a} \cdot \vec{b} = 0$ .
Orthogonality condition: For two vectors $\vec{a}, \vec{b}$ both distinct from $\vec{0}$ , we have: $\vec{a} \perp \vec{b} \Leftrightarrow \vec{a} \cdot \vec{b} = 0$
Scalar square: For every vector $\vec{a}$ , we have: $\vec{a}^2 = |\vec{a}|^2$
Cosine formula: If $\vec{a}, \vec{b}$ are two vectors distinct from $\vec{0}$ , then: $\cos(\vec{a}, \vec{b}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$

Properties of the Dot Product

Commutative property: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
Associative property with a scalar: $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b}) = \vec{a} \cdot (k\vec{b})$
Distributive property: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

Application in Physics

Mechanical work: If a constant force $\vec{F}$ acts on an object at point $M$ causing the object to undergo a displacement $\overrightarrow{MN}$ , then the work $A$ done by the force is calculated by: $A = \vec{F} \cdot \overrightarrow{MN}$ (where force $\vec{F}$ has magnitude measured in Newtons, displacement $MN$ is measured in meters, and work $A$ is measured in Joules).

Extensions: Centroid of a Triangle and Centroid of a Tetrahedron

Property of the centroid of a triangle: Similar to the plane, if $G$ is the centroid of triangle $ABC$ , then for any arbitrary point $O$ , we have: $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = 3\overrightarrow{OG}$
Centroid of a tetrahedron: Point $I$ is called the centroid of tetrahedron $ABCD$ if it satisfies: $\overrightarrow{IA} + \overrightarrow{IB} + \overrightarrow{IC} + \overrightarrow{ID} = \vec{0}$ (which is equivalent to $\overrightarrow{AI} = 3\overrightarrow{IG}$ , where $G$ is the centroid of triangle $BCD$ ).