AH Checklist (Unit 3) AH Checklist (Unit 3) Matrices

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1 AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow that the order of a matrix is give as m (read m by ), where m is the umber of rows ad the umber of colums ad is writte as: A ( a ) def m a a a a 3 a a a a 3 a a a a a a a a m m m3 m Kow that the mai diagoal (aka leadig diagoal ) of ay matrix is the set of etries a where i j Kow that the elemet i row i ad colum j of a matrix is writte as a ad called the (i, j) th etry of A Kow that a row matrix is a matrix ad is writte as: ( a a a a ( ) ) Kow that a colum matrix is a m matrix ad is writte as: a a a a ( m ) m Kow that a square matrix (of order m m or just of order m) is a matrix with the same umber of rows as colums (equal to m) ad is writte as: a a a a a a a a a m m m m mm Kow that the idetity matrix (of order m) is the m m matrix all of whose etries are 0 apart from those o the M Patel (August 0) St. Machar Academy

2 AH Checklist (Uit 3) AH Checklist (Uit 3) mai diagoal, where they all equal : I def m Kow that the zero matrix (of order m ) is the m matrix all of whose etries are 0: O def m Kow that the traspose of a m matrix A (deoted A ) is the m matrix obtaied by iterchagig the rows ad colums of A ((a ) deotes the (i, j ) th etry of A ): (a ) def a ji Obtai the traspose of ay matrix Kow that matrices are equal if they both have the same order ad all correspodig etries are equal Kow that matrices ca be added or subtracted oly if they have the same order Add or subtract matrices A ad B (thus obtaiig the matrix sum A + B or matrix differece A B ) by addig or subtractig the correspodig etries of each matrix (( a ± b deotes the (i, j ) th etry of the sum or differece): ( a b) ± def a ± b Add or subtract more tha matrices Multiply ay matrix A by a real scalar k (thus obtaiig the scalar multiplicatio ka ) by multiplyig each etry of A by k (( ka ) deotes the (i, j ) th etry of ka ): ) ( ka ) def ka (k R) Scalar multiply a matrix by a real scalar Kow, use ad verify the followig matrix properties (for k R): M Patel (August 0) St. Machar Academy

3 AH Checklist (Uit 3) AH Checklist (Uit 3) A + B B + A (A + B ) + C A + (B + C ) k (A + B ) ka + kb (A + B ) A + B ( A ) A ( ka ) ka Kow that matrices A ad B ca oly be multiplied i the order A times B (thus obtaiig the matrix product AB ) if the umber of colums of A equals the umber of rows of B Multiply matrices A (of order m ) ad B (of order p) accordig to the rule, where ( ab deotes the (i, j ) th etry of AB : ( ab ) def k a b ( i m ad j p ) ik kj Kow that, i geeral: A B B A Kow that whe formig AB, we say that B is pre-multiplied by A, or A is post-multiplied by B Multiply or more matrices where m ad are at most equal to 3 Kow that a (square) matrix A ca be multiplied by itself ay umber of times (thus obtaiig the th power of A ): A def times ) A A A A Kow, use ad verify the followig matrix properties: A (BC ) (AB ) C A (B + C ) AB + AC ( AB ) B A Kow that a matrix A is symmetric if A A (thus, A is square) Kow that a symmetric matrix is symmetrical about the mai diagoal M Patel (August 0) 3 St. Machar Academy

4 AH Checklist (Uit 3) AH Checklist (Uit 3) Kow that a matrix A is skew-symmetric (aka ati-symmetric) if A A (thus, A is square) Kow that a skew-symmetric matrix has all mai diagoal etries equal to 0 Kow that a square matrix A (of order ) is orthogoal if: A A Kow that a system of m equatios i variables may be writte i matrix form, where A is the m matrix of coefficiets (aka coefficiet matrix), x is the solutio vector ad b a m colum vector as: A x b Kow that for a matrix A, the mior of etry a is the determiat (deoted M ) of the ( ) ( ) matrix formed from A by deletig the i th row ad j th colum of A Kow that the cofactor of etry a is the quatity: I C def ( ) i + j Kow that for the determiat of a geeral matrix is give by the Laplace expasio formula : det (A ) A def j M a C (i,, 3,, ) Kow that a system of equatios i ukows has a solutio if the determiat of the coefficiet matrix is o-zero Kow that the determiat of a matrix is: A ( a ) a Kow that the determiat of a matrix is: A a c b d ad bc Calculate the determiat of ay matrix Kow that the determiat of a 3 3 matrix is: A a b c d e f g h i e a h f i d b g f i d + c g e h M Patel (August 0) 4 St. Machar Academy

5 AH Checklist (Uit 3) AH Checklist (Uit 3) Calculate the determiat of ay 3 3 matrix Kow that a matrix A has a iverse if there is a matrix (deoted A ) such that: AA I or A A Kow that there is oly iverse (if it exists) for ay give matrix Kow that a matrix is ivertible (aka o-sigular) if it has a iverse Kow that if a matrix does ot have a iverse, the it is said to be o-ivertible (aka sigular) Kow that a matrix is ivertible iff det (A ) 0 Kow that a matrix is o-ivertible iff det (A ) 0 Give a missig variable i a or 3 3 matrix, obtai the value(s) of this variable for which the matrix is sigular or o-sigular Kow that the cofactor matrix of square matrix A is the matrix C whose (i, j ) th etry is C Kow that the adjugate (aka classical adjoit) of a square matrix A is the traspose of the cofactor matrix C : adj (A ) def C Kow that the iverse of a matrix A is give by: A adj ( A ) det ( A ) Calculate the iverse of a x matrix usig the formula: A d ad bc c I b a Kow, use ad verify the followig matrix properties: AB A B ka k A (k R, N, A is ) A A A A M Patel (August 0) 5 St. Machar Academy

6 AH Checklist (Uit 3) AH Checklist (Uit 3) ( ) A A ( A ) ( A ) ( ka ) A k A B ( ) B A Calculate the iverse of a 3 x 3 matrix usig EROs by formig a big Augmeted matrix cosistig of A o the LHS ad the idetity matrix o the right, the row reducig the Augmeted Matrix util the idetity is reached o the left ad whatever remais o the RHS is A Kow that a system of equatios A x b ca be solved by premultiplyig each side of this equatio by A, ad so the solutio vector is obtaied as: x A b Kow that a liear trasformatio i the plae is a fuctio that seds a poit P(x, y) to a poit Q(ax + by, cx + dy) for a, b, c, d R Kow that a liear trasformatio i the plae ca be described as a matrix equatio: x y def ax + by cx + dy a c b x d y Kow that i the above equatio, the matrix with etries a, b, c ad d is called the trasformatio matrix Kow that if a trasformatio is represeted by a matrix A, the the reverse of that trasformatio is represeted by A Kow that a trasformatio matrix ca be obtaied by cosiderig the effect of a geometrical trasformatio o the poits (0, ) ad (, 0) Kow that a ivariat poit is oe that has the same image uder a trasformatio Fid ivariat poits of a give trasformatio by solvig the equatio: a c b x d y x y Kow that a reflectio i the lie y x M Patel (August 0) 6 St. Machar Academy

7 AH Checklist (Uit 3) AH Checklist (Uit 3) has trasformatio matrix: 0 0 Kow that a aticlockwise rotatio of agle θ (about the origi) has trasformatio matrix: cos θ si θ si θ cos θ Kow that a dilatatio (aka scalig) has trasformatio matrix: k 0 0 k (k R) Derive trasformatio matrices for other geometrical trasformatios, for example: Reflectio i the lie y x Reflectio i the x axis Reflectio i the y axis Half-tur aticlockwise about the origi Quarter-tur clockwise about the origi Kow that a combiatio of trasformatios is foud by matrix multiplicatio Derive the trasformatio matrix of a combiatio of trasformatios, ad describe the effect this combiatio has o a give poit, for example: reflectio i the x axis, the ati-clockwise rotatio of 30 ati-clockwise rotatio of π radias, the reflectio i the x axis elargemet of scale factor, the a clockwise rotatio of 60 Fid the equatio of the image of a give curve (possibly givig the aswer i implicit form) uder a give trasformatio M Patel (August 0) 7 St. Machar Academy

8 AH Checklist (Uit 3) AH Checklist (Uit 3) Vectors, Plaes ad Lies Skill Achieved? Kow that the directio ratio of a vector is the ratio of its compoets i the order x : y (: z) Determie the directio ratio of a vector Kow that vectors with the same directio ratio are parallel Kow that if α, β adγ are the agles the vector v makes with the x, y ad z axes, ad u is a uit vector i the directio of v, the cos α, cos β ad cosγ are the directio cosies of v ad: cos α u cos β cos α + cos β + cos γ cos γ Determie the directio cosies of a vector Kow that the vector product (aka cross product) of vectors a ad b, where θ is the agle from a to b, ad is a uit vector at right agles to both a ad b, is defied as: a x b def a b si θ Kow that the vector product is oly defied i 3D ad a x b is a vector at right agles to both a ad b Kow that the vector product is a vector, ot a scalar Kow that the 3 uit vectors, i, j ad k satisfy: i x j k j x k i k x i j ad i x i j x j k x k 0 Kow the followig properties of the vector product: a x a 0 a x b - b x a M Patel (August 0) 8 St. Machar Academy

9 AH Checklist (Uit 3) AH Checklist (Uit 3) a x (b + c) (a x b) + (a x c) (a + b) x c (a x c) + (b x c) Kow that if the vector product of vectors is 0, the they are parallel Kow that if o-zero vectors are parallel, the their vector product is 0 Give the compoets of vectors, calculate their vector product usig the compoet form of the vector product : a x b ( a b a b ) i + ( a b 3 3 a b ) j + ( a b 3 3 a b ) k Give the magitude of vectors ad the agle betwee them, calculate their vector product Give vectors i compoet form ad the agle betwee them, calculate their vector product Give vectors i compoet form, calculate their vector product Calculate the scalar triple product of 3 vectors usig: [a, b, c] a (b x c) def a a a 3 b b b 3 c c c 3 Kow that a Cartesia equatio of a plae cotaiig a poit P(x, y, z) is: ax + by + cz d (a, b, c, d, x, y, z R) Kow that the equatio of a plae is ot uique Determie whether or ot a poit lies o a plae Kow that a vector is parallel to a plae if it lies i the plae Kow that a ormal vector is at right agles to ay vector i the plae ad is give by: a b c Kow that parallel plaes have the same directio ratios for their ormals Kow that plaes are coicidet if they have the same equatio (possibly after simplifyig oe of the equatios) Kow that the agle betwee plaes is defied to be the acute agle betwee their ormals Give plaes, calculate the agle betwee them M Patel (August 0) 9 St. Machar Academy

10 AH Checklist (Uit 3) AH Checklist (Uit 3) Fid a equatio for a plae, give 3 poits i the plae Fid a equatio for a plae, give vectors i the plae Fid a equatio for a plae give poit o the plae ad a ormal to the plae Calculate the distace betwee plaes Kow that, for a poit A (with positio vector a) i a plae, ad vectors b ad c parallel to the plae, a vector equatio (aka parametric equatio) of a plae for a poit R (with positio vector r), where t ad u are real parameters, is: r a + t b + u c Fid a vector equatio for a plae Covert betwee the differet types of equatios for a plae Kow that a vector equatio for a lie i 3D with directio vector u a i + b j + c k passig through a poit A(x, y, z ) ad a geeral poit P(x, y, z) is: p a + t u (t R) Kow that the above equatio ca be writte i parametric form as: x x + at, y y + bt, z z + ct Kow that the above equatio ca be writte i symmetric form (aka stadard form or caoical form), provided that oe of a, b or c equal 0, as: x x y y z z t a b c Covert, where possible, betwee the 3 forms of a lie equatio Kow that a lie equatio is ot uique Verify that a poit lies o a lie Give poits o a lie, determie a equatio for L Kow that lies are parallel if the directio ratios of their directio vectors are equal Kow that lies are coicidet if they have the same equatio (possibly after simplifyig oe of the equatios) Give a poit lyig o a lie L ad a directio vector for L, determie a equatio for L Give a poit lyig o a lie L, ad vectors that are perpedicular to the directio of L, obtai the equatio of L Kow that o-coicidet lies i 3D may (i) be parallel (ii) itersect at a poit or (iii) be skew (either parallel or itersectig) M Patel (August 0) 0 St. Machar Academy

11 AH Checklist (Uit 3) AH Checklist (Uit 3) Give lies, determie whether or ot they itersect Give lies (either both i parametric form, both i symmetric form, or oe i parametric form ad the other i symmetric form) that itersect, fid their poit of itersectio Give the equatios of lies, calculate the size of the acute agle betwee them Calculate the shortest distace betwee straight lies Give a lie ad a poit o a plae that is perpedicular to the lie, determie a equatio of the plae Calculate the agle betwee a lie ad a plae Calculate the itersectio poit of a lie ad a plae Determie the shortest distace betwee a poit o a plae ad a lie perpedicular to a ormal of the plae Kow that o-coicidet plaes may either (i) ot itersect or (ii) itersect i a straight lie Give plaes, determie whether or ot they itersect Give itersectig plaes, fid their lie of itersectio Kow that the itersectio of 3 plaes ca be modelled by a system of 3 equatios i 3 variables ad the possibilities are: Uique itersectio poit lie of itersectio parallel lies of itersectio 3 parallel lies of itersectio plae of itersectio o itersectio Fid the itersectio of 3 plaes M Patel (August 0) St. Machar Academy

12 AH Checklist (Uit 3) AH Checklist (Uit 3) Further Sequeces ad Series Skill Achieved? Kow d Alembert s ratio test (aka ratio test) for decidig whether a series of positive terms, coverges or diverges: u, u + lim u u + lim u < series coverges > series diverges u + lim o coclusio u Kow that a series (ot ecessarily oe with all terms positive) u is absolutely coverget if u coverges Kow that ay absolutely coverget series is coverget Kow that if u u coverges but u diverges, the is said to be coditioally coverget Kow that ay rearragemet of a absolutely coverget series coverges to the same limit Kow the Riema rearragemet theorem, amely, that ay coditioally coverget series ca be rearraged to coverge to ay real umber, or rearraged to diverge Kow that for absolute covergece, d Alembert s ratio test is: u + lim < series coverges absolutely u u + lim > series diverges u M Patel (August 0) St. Machar Academy

13 AH Checklist (Uit 3) AH Checklist (Uit 3) u + lim o coclusio u Kow that the x values for which a power series coverges is called the iterval of covergece Kow that d Alembert s ratio test for a power series a x is: 0 a x + lim < a 0 a x coverges absolutely a x + lim > a a x diverges 0 a x + lim o coclusio a Fid S for power series (statig the iterval of covergece), for example: S + 3x + 5x + 7x ( ) x + S + 3x + 7x + 5x 3 + Kow that the Maclauri Series (aka Maclauri Expasio) of a fuctio f is the power series: f (x) 0 ( ) f (0) x! Kow that a trucated Maclauri series ca be viewed as a approximatio of a fuctio by a polyomial Write dow or fid the Maclauri series for: e x si x cos x ta x l ( + x) M Patel (August 0) 3 St. Machar Academy

14 AH Checklist (Uit 3) AH Checklist (Uit 3) Kow the rage of validity of the above 5 Maclauri series Give the Maclauri series for f (x), obtai the Maclauri series for f (ax) (a R {0}), for example: e x si 3x Evaluate Maclauri series that are a combiatio of the above, for example: ( + x) l ( + x) (first 4 terms) l (cos x) (up to the term i x 4 ) si x (up to the term i x 4 ) cos x (up to the term i x 4 ) e x si x (first 3 o-zero terms) e x x + (up to the term i x 4 ) cos x (up to the term i x 4 ) cos 6x (first 3 o-zero terms) x l ( + x) x l ( x) (first 3 o-zero terms) (first 3 o-zero terms) x l (4 x ) (first o-zero terms) x + 6 x 4 x x ( + ) ( 4) (first o-zero terms) + si x (first 3 o-zero terms) + x + x (first 4 terms) ( )( ) Kow, or derive, the biomial series (which is a Maclauri series): M Patel (August 0) 4 St. Machar Academy

15 AH Checklist (Uit 3) AH Checklist (Uit 3) ( + x ) r k 0 r x k k (r R) Use Maclauri series to approximate values, for example: si e cos l Kow that a recurrece relatio is sometimes kow as a iterative scheme Kow that a iterative sequece is a sequece geerated by a iterative scheme Kow that each x is called a iterate Calculate iterates give a iterative scheme Kow that a fixed poit (aka coverget or limit) of a iterative scheme is a poit a satisfyig: a F (a) Fid fixed poits of a recurrece relatio, for example: x + x + 7 x x + x + x Use a iterative scheme of the form x F ( x ) to solve a equatio f (x) 0, where x F (x) is a rearragemet of the origial equatio Kow that cobweb ad staircase diagrams ca be used to illustrate covergece (or divergece) of a iterative scheme df Kow that a iterative scheme has a fixed poit if <, where the derivative is evaluated at ay poit x i a small regio of the fixed poit M Patel (August 0) 5 St. Machar Academy

16 AH Checklist (Uit 3) AH Checklist (Uit 3) Further Differetial Equatios Skill Achieved? Kow that a th order ordiary differetial equatio (ODE) is a equatio cotaiig (i) a fuctio of a sigle variable ad (ii) its derivatives up to the th derivative Kow that a liear ODE is oe of the form: (r) a ( x ) f (x) r ( r ) r 0 Kow that a th order liear ODE with costat coefficiets is of the above form, but where all the a are costat r Kow that if f (x) 0, the DE is called homogeeous, whereas if f (x) 0, the DE is called o-homogeeous (or ihomogeeous) Kow that a solutio of a DE is a fuctio that satisfies the DE Kow that solutios of a DE are of types: Geeral solutio Particular solutios Kow that the geeral solutio of a DE has arbitrary costats, whereas a particular solutio has o arbitrary costats (as they are evaluated usig iitial coditios) Kow that the solutios of a homogeeous equatio are called complemetary fuctios (CF) Kow that a st order liear ODE ca be writte i the form: dy + P (x) y f (x) Kow that to solve a st order liear ODE, the first step is to multiply the equatio as writte above by the itegratig factor : e P ( x ) Solve st order liear ODEs usig the itegratig factor method, for example: dy + y x x M Patel (August 0) 6 St. Machar Academy

17 AH Checklist (Uit 3) AH Checklist (Uit 3) x dy 3y x 4 (x + ) dy 3y (x + ) 4 Fid a particular solutio of a differetial equatio that is solved usig the itegratig factor method Kow that a d order liear ODE with costat coefficiets ca be writte i the form (ihomogeeous equatio): a + b dy + c y f (x) Kow that the auxiliary equatio (aka characteristic equatio) of the above DE is: a p + b p + c 0 (p C) Kow that the CF of the above DE ca be writte i oe of 3 forms depedig o the ature of the roots of the auxiliary equatio Kow that if the auxiliary equatio has real (distict) roots p ad p, the the CF is of the form: y A e p x + B e p x (A, B R) CF Kow that if the auxiliary equatio has real (repeated) root p, the the CF is of the form: y ( A + Bx ) e px (A, B R) CF Kow that if the auxiliary equatio has a pair of complex (cojugate) roots p r + is ad p r is, the the CF is of the form: y e rx ( A cos sx + B si sx ) (A, B R) CF Kow that a particular itegral (PI) is a solutio of the ihomogeeous equatio ad is chose to be of a similar form as the fuctio f (x) : f (x) C x + D y S x + PI f (x) C x + D x + E y S x + x + U PI f (x) C e px y S e px PI M Patel (August 0) 7 St. Machar Academy

18 AH Checklist (Uit 3) AH Checklist (Uit 3) f (x) C si px y S si px + cos px PI f (x) C cos px y S si px + cos px PI Kow slight variatios of the last 3 PIs with a additioal costat: f (x) C e px + D y S e px + PI f (x) C si px + D y S si px + cos px + U PI f (x) C cos px + D y S si px + cos px + U PI Kow that if f (x) is of the same form as a term i the CF, try the same form as xf (x) for the PI; if this is of the same form as a term i the CF, try the same form asx f (x) for the PI Kow that the geeral solutio of the ihomogeeous equatio is: y G y + y CF PI Fid the geeral solutio of d order liear ODEs with costat coefficiets, for example: + dy + dy 3y 6x + 5y 4 cos x 4 dy + 4y e x 3 dy + y 0 si x + dy + y dy + 9y e x 3 dy + y x M Patel (August 0) 8 St. Machar Academy

19 AH Checklist (Uit 3) AH Checklist (Uit 3) + 4 dy + 5y 0 dy y ex + 5 dy 4y 3 si x + 4 cos x + dy y e x Fid the particular solutio of a d order liear ODE with costat coefficiets give iitial coditios, such as: dy y ex + give that y ad dy whe x 0. M Patel (August 0) 9 St. Machar Academy

20 AH Checklist (Uit 3) AH Checklist (Uit 3) Number heory ad Further Proof Methods Skill Achieved? Kow that whe A B, A is said to be sufficiet for B ad B is said to be ecessary for A Kow that i the bicoditioal A B, A is said to be ecessary ad sufficiet for B (ad vice versa) Prove statemets ivolvig fiite sums by iductio, for example: (3 ) (3 + ) ( N) 3 ( r r ) ( ) ( + ) ( N) r r( r + )( r + ) r 4 ( + )( + ) ( N) r( r + ) r ( ) + ( N) Prove statemets ivolvig matrices usig iductio, for example: A 0 A + ( N) A, B square matrices with AB BA A B BA ( N) Prove statemets ivolvig differetiatio usig iductio, for example: d x ( x e ) (x + ) e x ( N) Prove other statemets usig iductio, for example: > 3 ( 0) is divisible by 4 ( N)! > ( > 3) M Patel (August 0) 0 St. Machar Academy

21 AH Checklist (Uit 3) AH Checklist (Uit 3) is divisible by 5 ( ) Kow that the Divisio Algorithm states that, give a, b N,! q (quotiet), r (remaider) N satisfyig: a bq + r (0 r < b) Kow that the greatest commo divisor (GCD) (sometimes called the highest commo factor) of atural umbers is the biggest atural umber that exactly divides those umbers; the GCD of a ad b is deoted GCD(a, b) Kow that whe a bq + r, GCD(a, b) GCD(b, r) Kow that repeated applicatio of the Divisio Algorithm util the GCD is reached is called the Euclidea Algorithm Use the Euclidea Algorithm to fid the GCD of umbers Kow that GCD(a, b) ca be writte as GCD(a, b) ax + by Use the Euclidea Algorithm to write the GCD i the above form Kow that umbers are coprime (aka relatively prime) if their GCD Use the Euclidea Algorithm to fid itegers x a such that: 49 x + 39 y 3 x + 7 y 599 x + 53 y Kow that a liear Diophatie equatio is a equatio of the form: ax + by c (a, b, c, x, y Z) Kow that a liear Diophatie equatio of the above form has a solutio provided that GCD(a, b) divides c Kow that if a liear Diophatie equatio has a solutio, the it has ifiitely may solutios Solve liear Diophatie equatios usig the Euclidea Algorithm, for example: 4x + 3y 8 5x + 7y 4 5x + 7y Show that give liear Diophatie equatios do ot have ay solutios, for example: M Patel (August 0) St. Machar Academy

22 AH Checklist (Uit 3) AH Checklist (Uit 3) 4x + y 3 8x + 40y Kow that ay umber may be writte uiquely i base as: i k 0 r k i k i ( r r r r r 0 ) k k Kow that digits bigger tha 9 are give by the abbreviatios A 0, B, C, D 3, E 4 etc. Use the Divisio Algorithm to covert a umber i base 0 to aother base by obtaiig remaiders Covert a umber writte i o-base 0 to base 0, for example: (0) 3 (0) 0 (4E) 7 (8) 0 Covert a umber writte i base 0 to o-base 0, for example: (3) 0 (450) 7 (59) 0 (47) 3 Covert a umber from oe base to aother, either of which are base 0, by goig through base 0 first,for example: ( 0) 4 (73) 0 (8) 9 ( ) (666) 0 (556) (5D) 6 (93) 0 (333) 5 (5A) (70) 0 (55) 3 M Patel (August 0) St. Machar Academy

Mathematics 3 Outcome 1. Vectors (9/10 pers) Lesson, Outline, Approach etc. This is page number 13. produced for TeeJay Publishers by Tom Strang

Mathematics 3 Outcome 1. Vectors (9/10 pers) Lesson, Outline, Approach etc. This is page number 13. produced for TeeJay Publishers by Tom Strang Vectors (9/0 pers) Mathematics 3 Outcome / Revise positio vector, PQ = q p, commuicative, associative, zero vector, multiplicatio by a scalar k, compoets, magitude, uit vector, (i, j, ad k) as well as