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Important Points, , SCALAR OR DOT PRODUCT OF TWO VECTORS, @) The scalar or dot product of two non - zero vectors a and b, denoted by 'a.b' js, defined as @.b = |a|’|b | cos (a, b). This is a scalar. If either 4 = 0 (or) b =0, then, , we define a.b = 0. Ifwe write (a, b) =, then a. b =|a| |b | cos 0, ifa#0,b+0,, since 0 < (a, b) = @< 180°, we get, DEO 8 = 90° a. b S20;, , ii) @ = 90° > 4.b = Oand the vectors a and b are perpendicular., iii) 90° < @< 180° > 3.b <0, , iv) a.b=b.a, , V) a.(b+0)=a.b+ac, , vi) If a,b are parallel, 3.5 =+\a| |b., , vii) Ifl, me R, (la).(mb) =lm(a.b), , , , , , , , , , , , , , , , , , , , , , (2) i) Projection of b on @ (or) length of the projection of b on a = =, e (a.bya, ii) Orthogonal projection of b-on a = Gc a #0. _ (or), |, The projection vector of b on a = a a and its magnitude = —, iii) The component vector of b along a (or) parallel to 2 is ae, a, __ (a.b)B |, 3) Component vector of a along b = [BP component vector of a perpendicular to, |, ee, bese : |bP 2 |, a a ae, , Scanned with CamScanner

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ji) Equation of a plane passing through the origin and perpendicular to the unit vector 7, isr.n=0. |, , iii) Vector equation of a plane passing through a point A with position vector a and, perpendicular to a vector n is (r — a) ea Ob, , Perpendicular distance from the origin to the plane (r — a).n=0 is a. n, where 'a'is |, , the position vector of A in the plane and '1n' is a unit vector perpendicular to the plane., , Angle between two planes :, , If, and t, be two planes and M,, Mp are normals drawn to them, we define the angle |, , between M, and M, as the angle between 1, and t.,. If the angle between M, and M,, , M,.M,, , : fates fy |e, is @, the angle between the given planes ® = cos f My |IMo 1°, , Work done by a constant force F :, , i) Ifaconstant force F acting on a particle displaces it from a position 'A' to the position, B, then the work done 'W' by this constant force 'F is the dot product of the vectors, , representing the force F and displacement Ag, i.e., W = F.AB., , ii) IfFis the Poder of the forces F,, F,,......F,, then work doné in displacing the particle, from A to B is, W =F. AB+F). AB+......+F,. AB, , Cross Product or Vector Product of two vectors :, , The vector product or cross product of two non-parallel non — zero vectors 'a' and 'f ‘is, defined as axb =|a||b| sin @ f, where 'f'is a unit vector perpendicular to the plane |, , containing 'a' and 'b' such that a, 6 and 'f' form a vector triad in the right handed, system and (a, b) = ®, this is a vector, If either of a, b is azero vector or ‘a'is parallel |, to'b’, we define a x b = 0., , Some important results on vector product :, Deel xb |=|a||b |sine<|a||b| F, , |, |, |, |, |, , pm), , |axb |=|bxa|;, i) axb=-(bxa);, iit)’ “3x 2b asp, iv) (a)xb=ax(b)=-@xb), , y) laxmb=Im@xb) ;, , , , , , , , , , , , , , — . 108 TTT, Scanned with CamScanner

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TS OE SNS, , iv) If AB = a and AD = b are two adjacent sides of a parallelogram ABCD, then its, , vector area = axb and area = |axb | sq, units., , 1 :, v) Vector area of the quadrilateral ABCD = > (AC BD) and area of the quadrilatera|, , ABCD = 7 \ACxBD| sq. units., , Some useful formulas :, , i) If a, b are two non-zero and non-parallel vectors, then, ke a.a a., (axb)? = a°b?-a.b? =|. 5 5 5|, il) For any vector a, (axi)? +@x j)? +(axk)? = 2\a|?, iii) If a, b, © are the position vectors of the points A, B, C respectively, then the perpendicular, , |ACxAB| |(bxe)+(€xa)+(axb)|, , distance from c to the line AB is ay. = |b—a|, , Moment of a force :, , Let 0 be the point of reference (origin) and op =r be the position vector of a point p on, , the line of action of a force F. Then the moment of the force F about 0 is given by r x F., , Scalar triple product :, , let a, b, € be three vectors. We call (4 xb). € the scalar product of a, b and é. This, , is a scalar (real number). It is written as [a b Gl., i) If(@xb). € = 0, thenoneor more of the vectors a, b and @ should be Zero vectors., , !fa#0,b#0,c#0, then ¢ is perpendicular to a xb . Hence the vector c lies on the, , plane determined by 4 and h. Hence a, b and € are coplanar., it) Ifin a scalar triple product, any two vectors are parallel (equal), then the séalar triple, product is zeroi.e.,[a ab]=(a b b]=le b c]=0. :, iii) In a scalar triple product remains unaltered if the vectors are permutted cyclically, ie, lab C)=1b ¢ al=le a bl, |, However [4b €]=-[b 4] =-Ic b a] =-la © Bl., iv) Ina scalar triple product, the dot and cross are interchangeable, , ile, a-b XC =a xDne, , , , Scanned with CamScanner