8.3

Vectors in the Plane

Introduction

• A directed line segment has magnitude and direction.
• Equivalent directed line segments have the same magnitude and direction.
• A vector $v$ in the plane is the set of all directed line segments equivalent to a directed line segment $\vec{PQ}$, written as $v = \vec{PQ}$.
• A vector in standard position has its initial point at the origin and is called a position vector.

Component Form of a Vector

• A vector with initial point at the origin can be uniquely represented by the coordinates of its terminal point $(v<em>1, v</em>2)$.
• The component form of a vector is written as $v = \langle v<em>1, v</em>2 \rangle$, where $v<em>1$ and $v</em>2$ are the components of $v$.
• The component form of the vector with initial point $P(p<em>1, p</em>2)$ and terminal point $Q(q<em>1, q</em>2)$ is $\vec{PQ} = \langle q<em>1 - p</em>1, q<em>2 - p</em>2 \rangle$.
• The magnitude (or length) of $v$ is calculated by: $||v|| = \sqrt{v<em>1^2 + v</em>2^2}$.
• If $||v|| = 1$, then $v$ is a unit vector.
• $||v|| = 0$ if and only if $v$ is the zero vector, $0$.

Vector Operations

• Sum of vectors $u = \langle u<em>1, u</em>2 \rangle$ and $v = \langle v<em>1, v</em>2 \rangle$ is: $u + v = \langle u<em>1 + v</em>1, u<em>2 + v</em>2 \rangle$.
• Numbers in vector operations are called scalars.
• Scalar multiple of $k$ times $u = \langle u<em>1, u</em>2 \rangle$ is: $ku = \langle ku<em>1, ku</em>2 \rangle$.
• If $k$ is positive, $kv$ has the same direction as $v$, and if $k$ is negative, $kv$ has the opposite direction.

Properties of Vector Addition and Scalar Multiplication

• $u + v = v + u$
• $(u + v) + w = u + (v + w)$
• $u + 0 = u$
• $u + (-u) = 0$
• $c(du) = (cd)u$
• $(c + d)u = cu + du$
• $c(u + v) = cu + cv$
• $1(u) = u$
• $0(u) = 0$
• $||cv|| = |c|||v||$

Unit Vectors

• To find a unit vector $u$ that has the same direction as a given nonzero vector $v$, use $u = \frac{v}{||v||}$. This is called a unit vector.
• Standard unit vectors: $i = \langle 1, 0 \rangle$ and $j = \langle 0, 1 \rangle$.
• $v = \langle v<em>1, v</em>2 \rangle$ can be represented as $v = v<em>1i + v</em>2j$, where $v<em>1$ is the horizontal component and $v</em>2$ is the vertical component.
• The vector sum $v<em>1i + v</em>2j$ is a linear combination of the vectors $i$ and $j$.

Direction Angles

• If $u$ is a unit vector and $\theta$ is its direction angle, then $u = \langle x, y \rangle = \langle cos \theta, sin \theta \rangle$.
• If $v$ is any vector with direction angle $\theta$, then $v = \langle ||v||cos \theta, ||v||sin \theta \rangle$
• If $v = \langle a, b \rangle$, then the direction angle $\theta$ for $v$ can be determined from $tan \theta = \frac{b}{a}$.
• $\theta$ is the direction angle of vector $v$ where $v$ is in standard position and $\theta$ is the angle (measured counterclockwise) from the positive x-axis to $v$.

Applications

• Vectors can be used to model and solve real-life problems involving forces, velocities, and directions.