Study guide: Vector Algebra and Components

📋 Course Outline

1. Scalars and Vector Properties
2. Cartesian Coordinates and Components
3. Trigonometric Vector Resolution
4. Vector Magnitude and Direction
5. Unit Vectors and Component Notation

📖 1. Scalars and Vector Properties

🔑 Key Concepts & Definitions

• Scalar: A quantity defined by a single value that has magnitude but no direction; examples include mass, energy, power, temperature, speed, length, and time.
• Vector: A quantity with magnitude and direction (and sense), represented by an arrow; examples include velocity, force, moment, acceleration, and displacement.

📝 Essential Points

• The orientation of a vector's arrow is its line of action, and the arrowhead indicates the vector's sense along that line.

📌 A vector has a point of application only when it represents a force.

• A vector's magnitude is always a nonnegative number and may be denoted by its symbol or by absolute-value bars, such as $|A|$.

📌 Translating a vector anywhere in space without changing its orientation preserves its magnitude and direction, so the vector does not change.

💡 Memory Hook

Scalars carry magnitude alone; vectors carry magnitude and direction.

📖 2. Cartesian Coordinates and Components

🔑 Key Concepts & Definitions

• Cartesian coordinates: Represent a vector in a two-dimensional coordinate system as an ordered pair $A=(x,y)$.

📝 Essential Points

• In the Cartesian coordinate system, the positive and negative signs of the x- and y-coordinates determine the vector's quadrant.

• The vector with Cartesian coordinates $(13{,}7)$ has x-coordinate 13 and y-coordinate 7.

• The vectors $A=(3{,}6)$, $B=(-6{,}6)$, $C=(-3,-2)$, and $D=(11,-5)$ illustrate Cartesian coordinates in all four sign combinations.

📖 3. Trigonometric Vector Resolution

🔑 Key Concepts & Definitions

• Vector components: The x- and y-components of a vector are its projections onto the x- and y-axes, and the vector is their sum: $A=A_x+A_y$.

★ Must-know

📐 Formula - For a vector of magnitude $A$ at angle $\theta$ from the positive x-axis, $A_y=A\sin\theta$ and $A_x=A\cos\theta$.

Further detail

📐 Formula - For a right triangle formed by the vector and its components, $\sin\theta=\frac{A_y}{A}$, $\cos\theta=\frac{A_x}{A}$, and $\tan\theta=\frac{A_y}{A_x}$.

• The angle can be recovered from the components using $\theta=\sin^{-1}\frac{A_y}{A}$, $\theta=\cos^{-1}\frac{A_x}{A}$, or $\theta=\tan^{-1}\frac{A_y}{A_x}$.

• In a right triangle, the sine is opposite over hypotenuse, the cosine is adjacent over hypotenuse, and the tangent is opposite over adjacent (SOH–CAH–TOA).

📖 4. Vector Magnitude and Direction

★ Must-know

📐 Formula - For a vector with perpendicular components $A_x$ and $A_y$, its magnitude is $A=\sqrt{A_x^2+A_y^2}$.

Further detail

• For $A=(13{,}7)$, the magnitude is $\sqrt{13^2+7^2}=\sqrt{218}\approx14.76$.

• The direction angle of a vector can be determined from its component ratios using inverse trigonometric functions.

📖 5. Unit Vectors and Component Notation

🔑 Key Concepts & Definitions

• Unit vector: A vector with magnitude 1 and the same direction as the vector it represents.

★ Must-know

• The unit vectors along the coordinate axes are $\vec{i}$ for the x-axis, $\vec{j}$ for the y-axis, and $\vec{k}$ for the z-axis.

📐 Formula - A vector with components $v_x$ and $v_y$ can be written as $\vec{v}=v_x\vec{i}+v_y\vec{j}$.

Further detail

📐 Formula - A vector can be written as its magnitude multiplied by its unit vector: $A=A\,\vec{a}$.

• The vector with Cartesian coordinates $(4{,}3)$ is written in unit-vector form as $\vec{w}=4\vec{i}+3\vec{j}$.

• The vectors $(3{,}6)$, $(-6{,}6)$, $(-3,-2)$, and $(11,-5)$ correspond respectively to $3\vec{i}+6\vec{j}$, $-6\vec{i}+6\vec{j}$, $-3\vec{i}-2\vec{j}$, and $11\vec{i}-5\vec{j}$.