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 $P\left(x_1, y_1\right)$ and terminal point $Q\left(x_2, y_2\right)$, its component form is expressed as

 

• The magnitude of $\overrightarrow{PQ}$ 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 $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$

• If $\mathit{v}$ is a vector and , then $v=a\hat{i}+b\hat{j}$.

Vector Operations

Vector Addition:

• $u+v =\left(a\hat{i}+ b\hat{j}\right)+\left(c\hat{i}+ d\hat{j}\right))=\left(a+c\right)\hat{i}+ \left(b+d\right)\hat{j}$

Scalar Multiplication: The product of a vector  or $u=a\hat{i}+ b\hat{j}$ with a scalar $k$ is expressed as

• $ku=k\left(a\hat{i}+ b\hat{j}\right)=ka\hat{i}+ kb\hat{j}$

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 $\mathit{v}$ is expressed as

$\hat{u}=\frac{v}{||v||}$

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

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 $\mathit{v}$, then

• • The horizontal component of v is $||v||\cos \theta$.
  • The vertical component of v is $||v||\sin \theta$.

• The vector is represented as:

, or

$v=||v||\cos \theta \hat{i} + ||v||\sin \theta \hat{j}$

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 $\mathit{u}$, and $\mathit{v}$, then the dot product is given by:

$u·v=||u||||v|| \cos \alpha$

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 $\mathit{u}$ and $\mathit{v}$, then

$\alpha =CO{S}^{-1 }\left(\frac{u·v}{||u||||v||}\right)$

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

$\hat{u}=\frac{v}{||v||}$  

 

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

$\alpha ={\cos }^{-1} \left(\frac{1}{\sqrt{2}}\right)$ $=45°$

Cheat Sheet

• If a vector in XY-plane has an initial point $P\left(x_1, y_1\right)$ and the terminal point $Q\left(x_2, y_2\right)$, its component form is: .
• The magnitude of $\overrightarrow{PQ}$ 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 $\mathit{v}$ 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 $\overrightarrow{v}$ is a non-zero vector, then it's a unit vector $\hat{u}=\frac{v}{\left|v\right|}$.
• 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: $\theta ={\tan }^{-1} \left(\frac{v_2}{v_1}\right)$.
• The angle between two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ can be found by the formula: $\cos \theta =\frac{u}{\left|u\right|}·\frac{v}{\left|v\right|}$ 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 $u·v=0$