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# Precalculus - Vectors

## Introduction

• Vector quantities have a magnitude as well as direction.
• For example, velocity is a vector quantity as it includes speed (magnitude) and direction.

## Representation of a Vector

1. Component Form of Vector

• If a vector in XY-plane has initial point  and terminal point , its component form is expressed as

• The magnitude of $\stackrel{\to }{PQ}$ is

$||v||=\sqrt{{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({x}_{2}-{x}_{1}\right)}^{2}}$ $=\sqrt{{a}^{2}+{b}^{2}}$

2. Vector representation in terms of $\stackrel{\mathbf{^}}{\mathbf{i}}$ and $\stackrel{\mathbf{^}}{\mathbf{j}}$

• If $\mathbit{v}$ is a vector and , then $v=a\stackrel{^}{i}+b\stackrel{^}{j}$.

## Vector Operations

Scalar Multiplication: The product of a vector  or  with a scalar $k$ is expressed as

## Unit Vectors

• A vector whose magnitude is one is called a unit vector.
• In other terms, any non-zero vector divided by its magnitude is a unit vector.
• Mathematically, the unit vector of a vector $\mathbit{v}$ is expressed as

$\stackrel{^}{u}=\frac{v}{||v||}$

• A unit vector is in the direction of $\mathbit{v}$.

Note: The unit vector parallel to the x-axis is , while the unit vector parallel to the y-axis is .

## Horizontal & Vertical Components of a Vector

• If $\theta$ is the angle between the positive x-axis and a vector $\mathbit{v}$, then
• The horizontal component of v is .
• The vertical component of v is .
• The vector is represented as:

, or

## Dot Product of Vectors

• The dot product of two vectors , and  is given by:

$u·v=ac+bd$

• If $\alpha$ is the smallest non-negative angle between $\mathbit{u}$, and $\mathbit{v}$, then the dot product is given by:

Note: The dot product of two vectors is a real number, not a vector and hence it is also known as a scalar product.

## Angle Between Two Vectors

• If $\alpha$ is the smallest non-negative angle between $\mathbit{u}$ and $\mathbit{v}$, then

Note: Two non-zero vectors u and v are perpendicular if and only if $u·v=0$

## Solved Examples

Question 1: Given  and , find $u+v$.

Solution:

Question 2: Find , find $||v||$.

Solution: $||v||=\sqrt{{\left(3\right)}^{2}+{\left(-4\right)}^{2}}$ $=\sqrt{9+16}$ $=\sqrt{25}$ $=5$

Question 3: Find a unit vector in the direction .

Solution: $||v||=\sqrt{{\left(-5\right)}^{2}+{\left(12\right)}^{2}}$ $=\sqrt{25+144}$ $=\sqrt{169}$ $=13$

$\stackrel{^}{u}=\frac{v}{||v||}$

Question 4: Given $v=7i-8j$ and $w=3i-2j$, find $2v-9w$.

Solution: $2v-9w$$=2\left(7i-8j\right)-9\left(3i-2j\right)$ $=14i-16j-27i+18j$ $=-13i+2j$

Question 5: Find the dot product of $v=4i-j$ and $w=5i-3j$.

Solution: $v·w=4\left(5\right)+\left(-1\right)\left(-3\right)$ $=20+3=23$

Question 6: Find the measure of the smallest positive angle between the vectors $v=3i-2j$ and $w=i-5j$.

Solution:  $=\frac{\left(3i-2j\right)·\left(i-5j\right)}{\left(\sqrt{{3}^{2}+{\left(-2\right)}^{2}}\right)\left(\sqrt{{1}^{2}+{\left(-5\right)}^{2}}\right)}$ $=\frac{3+10}{\sqrt{13}\sqrt{26}}$ $=\frac{13}{13\sqrt{2}}$ $=\frac{1}{\sqrt{2}}$

$=45°$

## Cheat Sheet

• If a vector in XY-plane has an initial point  and the terminal point , its component form is: .
• The magnitude of $\stackrel{\to }{PQ}$ is: $\left|PQ\right|=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$.
• The component form of a vector $\mathbit{v}$ with a direction angle $\theta$ is .
• A unit vector has a magnitude of 1.
• Any non-zero vector divided by its magnitude is a unit vector. If $\stackrel{\to }{v}$ is a non-zero vector, then it's a unit vector $\stackrel{^}{u}=\frac{v}{\left|v\right|}$.
• The sum of two vectors  and  is expressed as .
• The product of a scalar $k$ with a vector  is .
• The direction angle of a vector  is given by: .
• The angle between two vectors $\stackrel{\to }{u}$ and $\stackrel{\to }{v}$ can be found by the formula:  where $u·v={u}_{x}·{v}_{x}+{u}_{x}·{v}_{x}$

## Blunder Areas

• The unit vector parallel to the x-axis is , while the unit vector parallel to the y-axis is .
• The dot product of two vectors is a real number, not a vector; hence, it is also known as a scalar product.
• Two non-zero vectors, u and v, are perpendicular if and only if $u·v=0$