SCALAR AND VECTOR QUANTITIES A scalar quantity has only magnitude and no direction. A vector quantity has both magnitude as well as direction. For example, velocity, force, acceleration, electric resistance, work, displacement, electric intensity, magnetic intensity; etc. are all vector quantities.
REPRESENTATION AND NOTATION OF VECTORS Generally a vector is determined by two points A, B such that magnitude of the vector is the length of the st. line AB and direction from A to B. The vector is written as . In vector AB, point A is called the origin and point B is AB called the terminus.
DIFFERENT TYPES OF VECTORS Unit Vectors : When vector is of unit magnitude, we call it a unit vector. If a be a vector whose magnitude is a or a , then the corresponding unit vector in that direction is ˆ a denoted by and its magnitude is unity. a ˆ ˆ Thus a a a or a . a Equal vectors : Two vectors are said to be equal if they have : i. same magnitude. ii. same direction. iii. independent of position of origin.
Like vectors : Two vectors are said to be like if they have the same direction in the same sense. Collinear vectors : Any number of vectors are said to be collinear when they are parallel to the same line regardless of their magnitudes. Here the term collinear is used as similar to parallel. Coplanar vectors : Three or more vectors are said to be coplanar when they are parallel to the same plane. Free vectors : Whenever the effect of a vector depends upon its magnitude and direction and is independent of the position of its origin, it is called a free vector.
Localized vector : When the position of origin is restricted to a certain specified point, then the vector is said to be a localized vector. Co-initial vectors : All such vectors, which have the same initial point, are called co-initial vectors. Null vectors : A vector having zero magnitude is known as null vector or zero vector. It is generally denoted by . Negative of 0 vector : The vector, which has same a magnitude as that of but the opposite direction, is called the negative of a and is denoted by . a a
Magnitude or modulus or length of a vector : If vector is represented in magnitude and direction by a then length AB is known as modulus or magnitude of AB a vector . a
TRIANGLE LAW OF VECTORS When two vectors a and b are represented in magnitude and direction by two sides ABand BC a of triangle ABC, taken in order, then the sum of these two vectors a represented in magnitude and direction is b by third side , taken ACreverse order. in The above result is known as triangle law of vectors.
PARALLELOGRAM LAW OF VECTORS When two vectors a and b are represented in magnitude and direction by the two adjacent sides and AB of a ||gm. ABCD, then the sum of these two vectors AD a b is represented in magnitude and direction by the diagonal of ||gm.
POSITION VECTOR OF A POINT Let O be the origin. We know that the position of any point P is specified uniquely by the vector OP, then vector OP , is known as position vector of P relative to O. Thus if , OP a,OQ b we know by triangle law of vectors OP PQ OQ PQ OQ OP b a
PROPERTIES OF VECTORS Vector addition is commutative. i.e. a b b a Vector addition is associative. i.e. a b c a b c ka ak k1 k 2 a k 2 k1 a k1 k 2 a k1a k 2 a k a b k a k b , where k, k1, k2 are positive scalars.
SOME RESULTS If a and b are like vectors, then each can be expressed as a scalar multiple of the other. Position vector of a point R dividing PQ internally in the mb na ratio m : n is c , m n where a, b are P. V. of P and Q respectively. The necessary and sufficient condition for three distinct points with position vectors to be b, c a, collinear is that there exist numbers x, y, z (not all zero), such that , where x + y + z=0. yb zc xa 0
If a, b, c are three non-zero non-coplanar vectors and x, y, z be three scalars such that xa yb z c, then x 0 0, y 0, z 0.
DOT PRODUCT (SCALAR PRODUCT) The dot product of two vectors aand bis defined as a .b a b cos , where is the angle between a and b. Properties of Dot Product : i. The dot product is commutative i.e. a .b b.aii. The dot product is associative with respect to scalar. i.e. k a .b k a.b a.k b iii. a 2 a .a a 2 ˆ2 i ˆ2 k 2 1 j ˆ
iv. If vectors a and bare perpendicular, then a .b . 0 a .bv. cos a b vi. a. b c a .b a .c vii. If a a1ˆ a 2 ˆ a 3 k, b b1ˆ b2 ˆ b3k i j ˆ i j ˆ a .b a.b a1b1 a 2 b2 a 3b3 cos a b a1b1 a 2 b 2 a 3 b3 = 2 2 2 2 2 2 a1 a 2 a 3 b1 b2 b3 viii. Two vectors a and b are perpendicular iff 2 2 2 a b a b
ix. Component of a vector r in the direction of r .a a 2 a a r .a and perpendicular to a r 2. a
VECTOR PRODUCT (OR CROSS PRODUCT) If a and b are two vectors and be the angle between them, then where ˆ ˆ a b , a b sin n is theunit vector n perpendicular to both and , such that ba form a ˆ right-handed system in anti-clockwise direction. a, b, n Properties of Cross Product :i. The cross product is not commutative i.e. ii. a b b a The cross product of two equal vectors is a null vector. i.e. iii. If and aare parallel, then a 0 . a biv. The cross product is associative a b to a scalar w.r.t 0 ka b a ka k a b
v. a a1 ˆ a 2 ˆ a 3k, b b1ˆ b2 ˆ b3k i j ˆ i j ˆ ˆ i ˆ j kˆ then a b a1 a2 a3 b1 b2 b3 vi. a b c a b a c vii. If a a1 ˆ a 2 ˆ a 3 k , i j ˆ b b1 ˆ b2 ˆ b3 k , i j ˆ 2 a b a 2 b3 a 3 b 2 then sin . a b 2 a1 2 b1
SCALAR TRIPLE PRODUCT If a, b, c be three vectors, then scalar triple product of three vectors, is the dot product of a b with vectors and c is written as : a b.c a, b, c Properties of Scalar Triple Product : i. a b c = Volume of parallelopiped with a, b, c edges. asii. etc. abc bca ca b
iii. abc ba c iv. abc 0 , if any two of the three vectors are collinear or equal. v. a b c 0 if a, b, c are coplanar.
VECTOR PRODUCT OF THREE VECTORS a b c a .c b b.c a i. a b, b c, c a 2 a b c and a b, b c, c a 0. 2ii. a b, b c, c a 2 a bc iii. If a, b, c are coplanar, then so are a b, b c, c a; a b, b c, c a; and a b, b c, c a .
SCALAR PRODUCT OF FOUR VECTORS If a, b, c, d be four vectors, then a b . c d is called product of four vectors. a .c a .d a b . c d b.c b.d The expression a b c d is called the vector product of four vectors a, b, c, d.
RECIPROCAL SYSTEM OF VECTORS If a, b, c be three non-zero, non-collinear and non- coplanar vectors, then the three vectors a , b , csuch that b c c a a b a ,b ,c a bc a bc a bc are called reciprocal system of vectors for vectors a, b, .c Properties : i. a.a b.b c.c 1 ii. a.b a.c b.a b.c c.a c.b 0
1iii. a b c abc iv. The vector a , b , c are non-zero, non collinear and non-coplanar. v. r r .a a r .b b r .c c = r .a a r .b b r .c c