If [math]a^{\rightarrow},b^{\rightarrow}[/math] and [math]c^{\rightarrow}[/math] be any three non-zero and non coplanar space vectors then any other vector [math]r^{\rightarrow}[/math] can be expressed uniquely as the sum of three space vectors parallel to the vectors [math]a^{\rightarrow},b^{\rightarrow}[/math] and [math]c^{\rightarrow}[/math] . [br][b]Proof: [/b]Let [math]OG^{^{\rightarrow}}=a^{\rightarrow}[/math],[math]OH^{\rightarrow}=b^{\rightarrow}[/math] and [math]OA^{\rightarrow}=c^{\rightarrow}[/math] be any three non-zero and non coplanar space vectors as shown in the figure. Let[math]OC^{\rightarrow}=r^{\rightarrow}[/math] be any other vector in the space. Join OE, OC and AC. Then by the parallelogram law of the addition of the vectors, we get [math]OE^{\rightarrow}[/math]= [math]OG^{\rightarrow}+OH^{\rightarrow}[/math]. Again [math]OC^{\rightarrow}=[/math][math]OA^{\rightarrow}+OE^{\rightarrow}[/math]. Hence [math]OC^{\rightarrow}=OG^{\rightarrow}+OH^{\rightarrow}+OA^{^{\rightarrow}}[/math] =[math]xa^{\rightarrow}+yb^{\rightarrow}+zc^{\rightarrow}[/math] or [math]r^{\rightarrow}=xa^{\rightarrow}+yb^{\rightarrow}+zc^{\rightarrow}[/math]..................(i)[br]For uniqueness, if possible,let [math]r^{\rightarrow}=x'a^{\rightarrow}+y'b^{\rightarrow}+z'c^{\rightarrow}[/math] ...............................(ii). Equating (i) and (ii) we get x =x ' , y = y' and z = z'. Hence the expression (i) is unique. Hence the theorem. [br][b]Note: [/b]See the related figure in the next resource page.[br][br][br][br][br]New Resources[list][*][url=https://www.geogebra.org/m/rkuhxwzq]Base X Machine (Addition with input)[/url][/*][*][url=https://www.geogebra.org/m/n8fyypvv]Alternate Segment Theorem[/url][/*][*][url=https://www.geogebra.org/m/tgfcyzw4]Point(s), a Line, Ray(s), Segment(s), & an angle - interactive & 3 questions[/url][/*][*][url=https://www.geogebra.org/m/jujhgnpa]Special Line through Triangle V1 (Theorem Discovery)[/url][/*][*][url=https://www.geogebra.org/m/wghztdeq]Six Pattern Blocks (14 each)[/url][/*][/list][br]Discover Resources[list][*][url=https://www.geogebra.org/m/bkh2p354]Line Reflections Demo[/url][/*][*][url=https://www.geogebra.org/m/tgmppp5b]Tech Assignment A part 1[/url][/*][*][url=https://www.geogebra.org/m/tf6cemzg]LineaireFormules[/url][/*][*][url=https://www.geogebra.org/m/gxsfxkff]Tut2, Q4[/url][/*][*][url=https://www.geogebra.org/m/tzcfwmd2]Geometry House Hunting[/url][/*][/list][br]Discover Topics[list][*][url=https://www.geogebra.org/t/frequency-distribution]Frequency Distribution[/url][/*][*][url=https://www.geogebra.org/t/polynomial-function]Polynomial Functions[/url][/*][*][url=https://www.geogebra.org/t/expected-value]Expected Value[/url][/*][*][url=https://www.geogebra.org/t/rhombus]Rhombus[/url][/*][*][url=https://www.geogebra.org/t/statistical-characteristics]Statistical Characteristics[/url][/*][/list][br][br]GeoGebra[list][*][url=https://www.geogebra.org/about]About[/url][/*][*][url=https://www.geogebra.org/team]Team[/url][/*][*][url=https://www.geogebra.org/newsfeed]News Feed[/url][/*][*][url=https://www.geogebra.org/partners]Partners[/url][/*][/list][br]Apps[list][*][url=https://www.geogebra.org/graphing]Graphing Calculator[/url][/*][*][url=https://www.geogebra.org/geometry]Geometry[/url][/*][*][url=https://www.geogebra.org/3d]3D Calculator[/url][/*][*][url=https://www.geogebra.org/download]App Downloads[/url][/*][/list][br]Resources[list][*][url=https://www.geogebra.org/materials]Classroom Resources[/url][/*][*][url=https://www.geogebra.org/groups]Groups[/url][/*][*][url=https://www.geogebra.org/a/14]Tutorials[/url][/*][*][url=https://help.geogebra.org/]Help[/url][/*][/list][br][list][*]Language: English[/*][/list][list][*][url=https://www.geogebra.org/tos]Terms of Service[/url] [url=https://www.geogebra.org/privacy]Privacy[/url] [url=https://www.geogebra.org/license]License[/url][/*][/list][list][*][url=https://www.facebook.com/geogebra]Facebook[/url] [url=https://www.twitter.com/geogebra]Twitter[/url] [url=https://www.youtube.com/user/GeoGebraChannel]YouTube[/url][/*][/list][br][br][br][br][br][br]

If [math]a^{\rightarrow},b^{\rightarrow}[/math] and [math]c^{\rightarrow}[/math] be any three non-zero and non coplanar space vectors then any other vector [math]r^{\rightarrow}[/math] can be expressed uniquely as the sum of three space vectors parallel to the vectors [math]a^{\rightarrow},b^{\rightarrow}[/math] and [math]c^{\rightarrow}[/math] . [br][b]Proof: [/b]Let [math]OG^{^{\rightarrow}}=a^{\rightarrow}[/math],[math]OH^{\rightarrow}=b^{\rightarrow}[/math] and [math]OA^{\rightarrow}=c^{\rightarrow}[/math] be any three non-zero and non coplanar space vectors as shown in the figure. Let[math]OC^{\rightarrow}=r^{\rightarrow}[/math] be any other vector in the space. Join OE, OC and AC. Then by the parallelogram law of the addition of the vectors, we get [math]OE^{\rightarrow}[/math]= [math]OG^{\rightarrow}+OH^{\rightarrow}[/math]. Again [math]OC^{\rightarrow}=[/math][math]OA^{\rightarrow}+OE^{\rightarrow}[/math]. Hence [math]OC^{\rightarrow}=OG^{\rightarrow}+OH^{\rightarrow}+OA^{^{\rightarrow}}[/math] =[math]xa^{\rightarrow}+yb^{\rightarrow}+zc^{\rightarrow}[/math] or [math]r^{\rightarrow}=xa^{\rightarrow}+yb^{\rightarrow}+zc^{\rightarrow}[/math]..................(i)[br]For uniqueness, if possible,let [math]r^{\rightarrow}=x'a^{\rightarrow}+y'b^{\rightarrow}+z'c^{\rightarrow}[/math] ...............................(ii). Equating (i) and (ii) we get x =x ' , y = y' and z = z'. Hence the expression (i) is unique. Hence the theorem.[br][br][br][br][br]New Resources[list][*][url=https://www.geogebra.org/m/rkuhxwzq]Base X Machine (Addition with input)[/url][/*][*][url=https://www.geogebra.org/m/n8fyypvv]Alternate Segment Theorem[/url][/*][*][url=https://www.geogebra.org/m/tgfcyzw4]Point(s), a Line, Ray(s), Segment(s), & an angle - interactive & 3 questions[/url][/*][*][url=https://www.geogebra.org/m/jujhgnpa]Special Line through Triangle V1 (Theorem Discovery)[/url][/*][*][url=https://www.geogebra.org/m/wghztdeq]Six Pattern Blocks (14 each)[/url][/*][/list][br]Discover Resources[list][*][url=https://www.geogebra.org/m/bkh2p354]Line Reflections Demo[/url][/*][*][url=https://www.geogebra.org/m/tgmppp5b]Tech Assignment A part 1[/url][/*][*][url=https://www.geogebra.org/m/tf6cemzg]LineaireFormules[/url][/*][*][url=https://www.geogebra.org/m/gxsfxkff]Tut2, Q4[/url][/*][*][url=https://www.geogebra.org/m/tzcfwmd2]Geometry House Hunting[/url][/*][/list][br]Discover Topics[list][*][url=https://www.geogebra.org/t/frequency-distribution]Frequency Distribution[/url][/*][*][url=https://www.geogebra.org/t/polynomial-function]Polynomial Functions[/url][/*][*][url=https://www.geogebra.org/t/expected-value]Expected Value[/url][/*][*][url=https://www.geogebra.org/t/rhombus]Rhombus[/url][/*][*][url=https://www.geogebra.org/t/statistical-characteristics]Statistical Characteristics[/url][/*][/list][br][br]GeoGebra[list][*][url=https://www.geogebra.org/about]About[/url][/*][*][url=https://www.geogebra.org/team]Team[/url][/*][*][url=https://www.geogebra.org/newsfeed]News Feed[/url][/*][*][url=https://www.geogebra.org/partners]Partners[/url][/*][/list][br]Apps[list][*][url=https://www.geogebra.org/graphing]Graphing Calculator[/url][/*][*][url=https://www.geogebra.org/geometry]Geometry[/url][/*][*][url=https://www.geogebra.org/3d]3D Calculator[/url][/*][*][url=https://www.geogebra.org/download]App Downloads[/url][/*][/list][br]Resources[list][*][url=https://www.geogebra.org/materials]Classroom Resources[/url][/*][*][url=https://www.geogebra.org/groups]Groups[/url][/*][*][url=https://www.geogebra.org/a/14]Tutorials[/url][/*][*][url=https://help.geogebra.org/]Help[/url][/*][/list][br][list][*]Language: English[/*][/list][list][*][url=https://www.geogebra.org/tos]Terms of Service[/url] [url=https://www.geogebra.org/privacy]Privacy[/url] [url=https://www.geogebra.org/license]License[/url][/*][/list][list][*][url=https://www.facebook.com/geogebra]Facebook[/url] [url=https://www.twitter.com/geogebra]Twitter[/url] [url=https://www.youtube.com/user/GeoGebraChannel]YouTube[/url][/*][/list][br][br][br][br][br][br]