Lernzettel: Vectors in Two Dimensions

📋 Course Outline

1. Scalars and Vectors
2. Vector Representation and Classification
3. Magnitude and Direction
4. Vector Addition and Subtraction
5. Scalar Multiplication
6. Position Vectors and Components
7. Applications of Plane Vectors

📖 1. Scalars and Vectors

🔑 Key Concepts & Definitions

• Scalar quantity : Fully described by its magnitude or numerical value alone.
• Vector quantity : Fully described by both a magnitude and a direction.

★ Must-know

📌 Distance is the scalar length traveled, whereas displacement is directed distance with a specified direction.

Further detail

• Mass, density, temperature, distance, speed, area, height, and rainfall amount are scalar quantities, whereas force, acceleration, velocity, displacement, and weight are vector quantities.

📖 2. Vector Representation and Classification

🔑 Key Concepts & Definitions

• Directed line segment : A vector is geometrically represented by a directed line segment, with its initial point at the tail and its terminal point at the arrowhead.
• Equal vectors : Two vectors are equal if and only if they have the same magnitude and the same direction.
• Column vector : A two-dimensional vector can be written as $\mathbf{a}=\begin{pmatrix}x\\y\end{pmatrix}$, where $x$ and $y$ are its horizontal and vertical components.
• Opposite vector : The opposite of a vector has the same magnitude as the original vector but the opposite direction.
• Parallel vectors : Vectors are parallel when they have the same direction or opposite directions.

💡 Memory Hook

Imagine an arrow from an initial point to a terminal point

📖 3. Magnitude and Direction

★ Must-know

📐 Formula — For $\mathbf{a}=\begin{pmatrix}x\\y\end{pmatrix}$, its magnitude is $|\mathbf{a}|=\sqrt{x^2+y^2}$.

📐 Formula — For points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$, the magnitude of $\overrightarrow{PQ}$ is $|\overrightarrow{PQ}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

• The direction of a plane vector is given by the angle it makes with a reference direction, commonly the horizontal or vertical axis.

Further detail

• A bearing-style direction is stated in the order north or south, angle measure, and east or west, such as N 53° E.

💡 Memory Hook

Components → magnitude → reference angle → bearing

📖 4. Vector Addition and Subtraction

🔑 Key Concepts & Definitions

• Triangle law : States that if $\overrightarrow{AB}=\mathbf{u}$ and $\overrightarrow{BC}=\mathbf{v}$, then $\mathbf{u}+\mathbf{v}=\overrightarrow{AC}$.
• Parallelogram law : For two vectors with a common initial point, their sum is represented by the diagonal of the parallelogram formed by the two vectors.

📝 Essential Points

• Vector subtraction is treated as addition of the first vector and the negative of the second vector, so $\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})$.

📐 Formula — For $\mathbf{a}=\begin{pmatrix}p\\q\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}r\\s\end{pmatrix}$, $\mathbf{a}+\mathbf{b}=\begin{pmatrix}p+r\\q+s\end{pmatrix}$ and $\mathbf{a}-\mathbf{b}=\begin{pmatrix}p-r\\q-s\end{pmatrix}$.

💡 Memory Hook

Place head to tail, then join the starting point to the endpoint

📖 5. Scalar Multiplication

🔑 Key Concepts & Definitions

• Scalar multiple : For a real scalar $k$ and vector $\mathbf{a}$, the scalar multiple $k\mathbf{a}$ has magnitude $|k|$ times the magnitude of $\mathbf{a}$.

★ Must-know

📌 If $k>0$, $k\mathbf{a}$ has the same direction as $\mathbf{a}$; if $k<0$, it has the opposite direction.

Further detail

📌 If $|k|>1$ a scalar multiple enlarges a vector, while if $0<|k|<1$ it shortens the vector.

• Two vectors are parallel when one can be expressed as a scalar multiple of the other.

💡 Memory Hook

Positive keeps direction; negative reverses it

📖 6. Position Vectors and Components

🔑 Key Concepts & Definitions

• Position vector : A vector whose initial point is the origin.

★ Must-know

📐 Formula — A vector from $A=(x_1,y_1)$ to $B=(x_2,y_2)$ has position-vector components $\overrightarrow{AB}=\begin{pmatrix}x_2-x_1\\y_2-y_1\end{pmatrix}$.

Further detail

📐 Formula — Using the unit vectors $\mathbf{i}=\begin{pmatrix}1\\0\end{pmatrix}$ and $\mathbf{j}=\begin{pmatrix}0\\1\end{pmatrix}$, a vector with components $x$ and $y$ is $\mathbf{u}=x\mathbf{i}+y\mathbf{j}$.

• The position vector $\mathbf{u}=\begin{pmatrix}3\\4\end{pmatrix}$ has magnitude $5$ and can be written as $\mathbf{u}=3\mathbf{i}+4\mathbf{j}$.

📖 7. Applications of Plane Vectors

★ Must-know

• A boat traveling 8 km south and then 8 km west has displacement magnitude $8\sqrt{2}$ km in the direction S 45° W.

📐 Formula — For two perpendicular forces, the resultant magnitude satisfies $|\mathbf{R}|=\sqrt{|\mathbf{F}_1|^2+|\mathbf{F}_2|^2}$.

• To solve a two-dimensional vector application, represent each displacement or force by components, combine horizontal and vertical components, and then determine the resultant magnitude and direction.

Further detail

• Perpendicular forces of 10 N and $10\sqrt{3}$ N produce a resultant force of 20 N.

📊 Synthesis Tables

Scalar and Vector Quantities

Category	Magnitude	Direction	Examples
Scalar	Required	Not required	Mass, density, temperature, distance, speed
Vector	Required	Required	Force, acceleration, velocity, displacement, weight