Vectors TopicsA Level BasicsColumn vector representationFinding the magnitude of vectorFinding the position vector and direction vectorsBasic geometry including ratiosand proving a straight lineFinding vectors parallel to an axis or other vectorsBasic angle calculations (includes between axes and within a triangle)Given magnitudes or angles, find unknowns𝑥,𝑦and 𝑧axis vectorsVector Equation Of A Straight Line 𝑟=#𝑎𝑏𝑐'+𝜆*𝑑𝑒𝑓./!"#0=𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,7$%&8=𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛(𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑡𝑜)Given a point the line passes through and a parallel vector -find the vector equation of the lineGiven a point the line passes through and a line parallel to -find the vector equation of the lineGiven 2 points the line passes through -find the vector equation of the lineGenerating other points that lie on a line or generating parallel vectors when given a vector line equation3 Ways/Forms of writing a vector line equation1.Position plus lambda times direction2.Grouping the position and direction together 𝑟=*𝑎+𝜆𝑑𝑏+𝜆𝑒𝑐+𝜆𝑓.3.Parametric𝑥=𝑎+𝜆𝑏𝑦=𝑏+𝜆𝑒𝑧=𝑐+𝜆𝑓Given a vector line equation, find another line equation Understanding what vectors parallel to an axis mean: Parallel to 𝑥axis 7'((8, Parallel to 𝑦axis 7('(8, Parallelto𝑧axis9010:Showing a point lies on a lineShowing a point does not lie on a lineGiven a point lies on a line, find one or more unknownsFinding position vectors or unknowns based on knowing distances (this often comes up with circles and radii)Collinear –showing points are collinear or not collinearGiven collinear and find unknownCartesian Form Of A Straight Line+,!$=-,"%=.,#&Given a point the line passes through and a parallel vector -find the cartesian equation of the lineGiven a point the line passes through and a line parallel to -find the cartesian equation of the lineGiven 2 points the line passes through -find the cartesian equation of the lineConverting from vector line equation to cartesian line equation and vice versaConverting from vector cartesian equation to vector line equation and vice versaVector Equation Of A Plane/+-.0=/!"#0+𝜆/$%&0+𝜇7/018/!"#0=𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛, 7$%&8𝑎𝑛𝑑7/018=𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠(𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑡𝑜𝑡ℎ𝑒𝑝𝑙𝑎𝑛𝑒)and are not parallel to each otherGiven 3 points -find the vector equation of a planeGiven a contained line and another point -find the vector equation of a plane (Hint: Generate 2 points from the line)3 Ways/Forms of writing a vectorplane equation1.Position plus 𝜆times direction plug 𝜇times another direction2.Grouping the position and direction together 𝑟=*𝑎+𝜆𝑑+𝜇𝑟𝑏+𝜆𝑒+𝜇𝑠𝑐+𝜆𝑓+𝜇𝑡.3.Parametric𝑥=𝑎+𝜆𝑑+𝜇𝑟𝑦=𝑏+𝜆𝑒+𝜇𝑠𝑧=𝑐+𝜆𝑓+𝜇𝑡Showing a point lies on a planeShowing a point does not lie on a planeGenerating other points that lie on a plane or parallel vectors when given a vector plane equationGiven a point lies on a plane, find one or more unknownsShowing 3 points are or are not coplanarDetermining whether 4 points lie in the same planeTwo ways to define a line: •If given a point and direction•If given two points (subtract points for direction)Ultimately,we just need to have a point and 1 directionand we can easily find the equation of a line in either vector or cartesian formFour ways to define a plane: •3 points (not all on same line)•Line and point not on line•2 distinct intersecting lines•2 parallel linesUltimately,we just need to have a point in a plane and aperpendicular direction
Cartesian Equation Of A Plane𝑔𝑥+ℎ𝑦+𝑖𝑧=𝑑𝑑=𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑓𝑜𝑟𝑚𝑜𝑟𝑖𝑔𝑖𝑛𝑡𝑜𝑝𝑙𝑎𝑛𝑒, 7!"#8=𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑣𝑒𝑐𝑡𝑜𝑟(𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟𝑡𝑜𝑡ℎ𝑒𝑝𝑙𝑎𝑛𝑒𝑡ℎ𝑖𝑠𝑡𝑖𝑚𝑒)Note: we plug point into 9𝑥𝑦𝑧:to getdGiven a point and perpendicular direction -find the cartesian equation of a planeGiven 3 points -find the cartesian equation of a planeGivenaline and a point not on a line -find the cartesian equation of a planeGiven 2 parallel lines -find the cartesian equation of a planeGiven 2 distinct intersecting lines -find the cartesian equation of a planeShowing a point lies on a planeShowing a point does not lie on a planeGiven a point lies on a plane, find one or more unknownsConverting from vector plane equation to cartesian plane equationConverting from vector cartesian equation to vector plane equationScalar ProductProperties and formulaParallel and perpendicular implicationsUsing the formula to find angles and areasFinding the exact value of 𝑐𝑜𝑠𝜃Finding a perpendicular vectorTo another given vectorTo two given vectorsTo a line and another line (including the coordinate of the foot of perpendicular)IntersectionsPoint and a line (occurs at a point)2 lines (occurs at a point or not at all)2 planes (occurs as a line or not at all)Line and plane (occurs at a point, line or not at all)3 planes (occurs at a point, line or not at all)At a pointAta lineNot at allPlane with axes Angle Between2 vectorsIn a triangle 2 linesLine and plane2 planesA line and its reflectionDistancesA point and a line2 linesA point and a plane2 parallel planesA plane and a parallel lineReflectionsPoint in a linePoint in a planeLine in a plane
VectorsFormulae SheetNotations 𝑣𝑒𝑐𝑡𝑜𝑟=𝒂, 𝑎, 𝑶𝑨CCCCCC⃗distance= OAVector Form𝑎𝒊+𝑏𝒋+𝑐𝒌≡I𝑎𝑏𝑐JProperties(addition/subtraction, multiplication and scalar product)I𝑎𝑏𝑐J±#𝑑𝑒𝑓'=#𝑎±𝑑𝑏±𝑒𝑐±𝑓'and 𝜆I𝑎𝑏𝑐J=#𝜆𝑎𝜆𝑏𝜆𝑐'I𝑎𝑏𝑐J.#𝑑𝑒𝑓'=𝑎𝑑+𝑏𝑒+𝑐𝑓Magnitude of a vectorNotation is ||NI𝑎𝑏𝑐JN=O𝑎2+𝑏2+𝑐2Unit VectorUnit vector of /!"#0='√!!4"!4#!7!"𝑐8Parallel and Perpendicular toParallel means vectors are a multiple of each otherPerpendicular means scalar product equals zeroAngle Between 2 vectorsAlways use the direction vectors𝜃=cos,'⎝⎜⎜⎛I𝑎𝑏𝑐J.#𝑑𝑒𝑓'NI𝑎𝑏𝑐JNW#𝑑𝑒𝑓'W⎠⎟⎟⎞Vector Equation of a lineTo find this we need:Point and direction(if given 2 points find the directions and use either point)𝑟=#𝑎𝑏𝑐'+𝜆*𝑑𝑒𝑓./!"#0=𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,7$%&8𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛(𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑡𝑜)Cartesian Equation of a line𝑥−𝑎𝑑=𝑦−𝑏𝑒=𝑧−𝑐𝑓/!"#0=𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,7$%&8𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛(𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑡𝑜)Parametric Form of a line𝑥=𝑎+𝜆𝑑,𝑦=𝑏+𝜆𝑒,𝑧=𝑐+𝜆𝑓Equation of a plane𝒓.𝒏=I𝑎𝑏𝑐J.𝒏where n is the normal vectorVector Equation of a planeTo find this we need: a point in plane and perp direction. If not given perp direction take the cross product of 2 direction vectors. Remember to find a direction we subtract 2 position vectors/+-.0=/!"#0+𝜆/$%&0+𝜇7/018/!"#0=𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛7$%&8𝑎𝑛𝑑7/018=𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠(𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑡𝑜)Cartesian Equation of a plane𝑎𝑥+𝑏𝑦+𝑐𝑧=𝑑𝑑=𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑓𝑜𝑟𝑚𝑜𝑟𝑖𝑔𝑖𝑛𝑡𝑜𝑝𝑙𝑎𝑛𝑒#𝑎𝑏𝑐'=𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑣𝑒𝑐𝑡𝑜𝑟(𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟𝑡𝑜)Scalar ProductNote: 𝜃is the angle between I𝑎𝑏𝑐Jand #𝑑𝑒𝑓'I𝑎𝑏𝑐J.#𝑑𝑒𝑓'=NI𝑎𝑏𝑐JNW#𝑑𝑒𝑓'W𝑐𝑜𝑠𝜃Vector ProductNote: 𝜃is the angle between I𝑎𝑏𝑐Jand #𝑑𝑒𝑓'I𝑎𝑏𝑐J×#𝑑𝑒𝑓'=#𝑏𝑓−𝑒𝑐−(𝑎𝑓−𝑐𝑑)𝑎𝑒−𝑏𝑑'orWI𝑎𝑏𝑐J×#𝑑𝑒𝑓'W=NI𝑎𝑏𝑐JNW#𝑑𝑒𝑓'Wsin𝜃Area of a Parallelogram𝐴=WI𝑎𝑏𝑐J×#𝑑𝑒𝑓'WI𝑎𝑏𝑐Jand #𝑑𝑒𝑓'form 2 adjacent sides of a parallelogramPerp Distance between point and planefrom (𝛼,𝛽,𝛾)to 𝑎𝑥+𝑏𝑦+𝑐𝑧=𝑑|𝑎(𝛼)+𝑏(𝛽)+𝑐(𝛾)+𝑑|√𝑎2+𝑏2+𝑐2Scalar Product Properties 0.𝒂=𝒂𝒂.𝒃=𝒃.𝒂(−𝒂).𝒃=−(𝒂.𝒃)(𝑘𝒂).𝒃=𝑘(𝒂.𝒃)𝒂.(𝒃+𝒄)=𝒂.𝒃+𝒂.𝒄If a and b are parallel: 𝒂.𝒃=|𝒂||𝒃|, moreover 𝒂.𝒂=|𝒂|2Cross Product Properties(optional)𝐚×𝒂=0𝒂×0=0×𝒂=0𝜆(𝒂×𝒃)=(𝜆𝒂)×𝒃=𝒂×(𝜆𝒃)𝒂×(𝒃+𝒄)=(𝒂×𝒃)+(𝒂×𝒄)𝒂×𝒃=−(𝒃×𝒂)𝒃.(𝒄×𝒂)=𝒄.(𝒂×𝒃)
