Introduction to Vectors

Unit Vector

• The formula for a unit vector $ extbf{u}$ in the direction of a non-zero vector $ extbf{a}$ is: $ extbf{u} = frac{ extbf{a}}{| extbf{a}|}$

Vector Equation of a Line

• The vector equation of a line passing through a point with position vector $ extbf{a}$ and parallel to vector $ extbf{b}$ is: $ extbf{r} = extbf{a} + t extbf{b}$ where $t$ is a scalar parameter

General Equation of a Plane

• The general equation of a plane in three-dimensional space is: $Ax + By + Cz + D = 0$, where $A$, $B$, $C$, and $D$ are constants, and $A$, $B$, and $C$ are not all zero.

Plane using a Point and Normal Vector

• The equation of a plane using a point with position vector $ extbf{a}$ and a normal vector $ extbf{n}$ is: $ extbf{n} · ( extbf{r} - extbf{a}) = 0$.

Applications of Vectors

• Areas where vectors are commonly applied include physics, engineering, computer graphics, and game development.

Vector vs. Scalar

• A vector has both magnitude (size) and direction, while a scalar has only magnitude.

Vector Addition

• Vector addition is performed by adding the corresponding components of the vectors.

Position Vector

• A position vector represents the location of a point in space relative to an origin.

Magnitude of a Vector

• The magnitude of the vector $ egin{pmatrix} 3 \ 4 \ \end{pmatrix}$ is $sqrt{3^2 + 4^2} = 5$.

Unit Vector Properties

• A unit vector has a magnitude of 1.

Scalar Multiplication

• When a vector is multiplied by a scalar, the magnitude of the vector is scaled by the scalar, and the direction is either unchanged (positive scalar) or reversed (negative scalar).

Vector Subtraction

• Subtracting the vector $ egin{pmatrix} 5 \ 3 \ \end{pmatrix}$ from the vector $ egin{pmatrix} 2 \ 1 \ \end{pmatrix}$ gives the resultant vector: $ egin{pmatrix} 2 \ 1 \ \end{pmatrix} - egin{pmatrix} 5 \ 3 \ \end{pmatrix} = egin{pmatrix} -3 \ -2 \ \end{pmatrix}$.

Three-Dimensional Vector Representation

• The vector $ egin{pmatrix} 1 \ 2 \end{pmatrix}$ would be represented in three-dimensional space as $ egin{pmatrix} 1 \ 2 \ 0 \ \end{pmatrix}$.

This quiz covers the fundamental concepts of vectors, including their definitions, representations, and operations such as addition and subtraction. Understand how vectors differ from scalars and explore graphical representations of these quantities. Perfect for beginners in vector mathematics.