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 Px1, y1 and terminal point Qx2, y2, its component form is expressed as

PQ=x2-x1, y2-y1 =a, b=v

  • The magnitude of PQ is

v=y2-y12+x2-x12 =a2+b2

 

2. Vector representation in terms of i^ and j^

  • If v is a vector and v=a, b, then v=ai^+bj^.

Vector Operations

Vector Addition:

  • u+v =a, b+c, d=a+c, b+d
  • u+v =ai^+ bj^+ci^+ dj^)=a+ci^+ b+dj^

Scalar Multiplication: The product of a vector u=a, b or u=ai^+ bj^ with a scalar k is expressed as

  • ku=ka, b=ka, kb
  • ku=kai^+ bj^=kai^+ kbj^

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 v is expressed as

u^=vv

  • A unit vector is in the direction of v.

Note: The unit vector parallel to the x-axis is i^=1, 0, while the unit vector parallel to the y-axis is j^=0, 1.

Horizontal & Vertical Components of a Vector

  • If θ is the angle between the positive x-axis and a vector v, then
    • The horizontal component of v is vcos θ.
    • The vertical component of v is vsin θ.
  • The vector is represented as:

v=vcos θ, vsin θ, or

v=vcos θ i^ + vsin θ j^

Dot Product of Vectors

  • The dot product of two vectors u=a, b, and v=c, d is given by:

u·v=ac+bd

  • If α is the smallest non-negative angle between u, and v, then the dot product is given by:

u·v=uv cos α

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 α is the smallest non-negative angle between u and v, then

α=COS-1 u·vuv

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

Solved Examples

Question 1: Given u=3, -5 and v=7, 1, find u+v.

Solution: u+v=3, -5+7, 1=3+7, -5+1 =10, -4

 

Question 2: Find v=3, -4, find v.

Solution: v=32+-42 =9+16 =25 =5

 

Question 3: Find a unit vector in the direction v=-5, 12.

Solution: v=-52+122 =25+144 =169 =13

u^=vv u^=-5, 1213 u^=-513, 1213

 

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

Solution: 2v-9w=27i-8j-93i-2j =14i-16j-27i+18j =-13i+2j

 

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

Solution: v·w=45+-1-3 =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 α= v·wvw =3i-2j·i-5j32+-2212+-52 =3+101326 =13132 =12

α=cos-1 12 =45°

Cheat Sheet

  • If a vector in XY-plane has an initial point Px1, y1 and the terminal point Qx2, y2, its component form is: PQ=x2-x1, y2-y1.
  • The magnitude of PQ is: PQ=x2-x12+y2-y12.
  • The component form of a vector v with a direction angle θ is v=vcos θ, vsin θ.
  • A unit vector has a magnitude of 1.
  • Any non-zero vector divided by its magnitude is a unit vector. If v is a non-zero vector, then it's a unit vector u^=vv.
  • The sum of two vectors a=a1, a2 and b=b1, b2 is expressed as a+b=a1+b1, a2+b2.
  • The product of a scalar k with a vector v=v1, v2 is kv=kv1, v2=kv1, kv2.
  • The direction angle of a vector v=v1, v2 is given by: θ=tan-1 v2v1.
  • The angle between two vectors u and v can be found by the formula: cos θ=uu·vv where u·v=ux·vx+ux·vx

Blunder Areas

  • The unit vector parallel to the x-axis is i^=1, 0, while the unit vector parallel to the y-axis is j^=0, 1.
  • 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