Introduction to Vectors

What is the formula for a unit vector $ extbf{u}$ in the direction of a non-zero vector $ extbf{a}$?

Which of the following represents the vector equation of a line passing through a point with position vector $ extbf{a}$ and parallel to vector $ extbf{b}$?

What is the general equation of a plane in three-dimensional space?

Which equation represents a plane using a point with position vector $ extbf{a}$ and a normal vector $ extbf{n}$?

Which of the following areas is commonly associated with the applications of vectors?

What distinguishes a vector from a scalar?

How is vector addition performed?

What is a position vector?

What is the magnitude of the vector $egin{pmatrix} 3 \ 4 \ \end{pmatrix}$?

Which statement about unit vectors is true?

What happens when a vector is multiplied by a scalar?

When subtracting the vector $egin{pmatrix} 5 \ 3 \ \end{pmatrix}$ from the vector $egin{pmatrix} 2 \ 1 \ \end{pmatrix}$, what is the resultant vector?

How would the vector $egin{pmatrix} 1 \ 2 \end{pmatrix}$ be represented in three-dimensional space?

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.