VECTORS
Physical qantities are of two types: scalers and vectors.
Scaler Quantity : Scaler quantities are those physical quantities which have only magnitude and no-direction such as mass, area, volume, temperature etc.
Vector Quantity : Vector quantities are those physical quantities which have both magnitude and direction such as velocity, force, displacement etc.
Modulus of a vector : The magnitude of a vector is called its modulus. The
length of the line and arrow express the modulus of a vector.
The two vectors are equal if they have the same modulus, direction and line. A vector is denoted as a (by marking an arrow) or by bold letter a.
DIFFERENT TYPES OF VECTORS
- Null vector : This is also called a zero If the initial and terminal points of a vector coincide, it is known as null vector. It is denoted by 0. It has a zero magni- tude.
- Unit vector : A vector having a magnitude of unit length is called a unit vector in the direction of This is denoted by a.
mod |a| = 1.
- Reciprocal vector : A vector having same direction as that of a given vector a but its magnitude is the reciprocal of the magnitude of the given vector a is called the reciprocal of a and is denoted by a-1.
1
Thus if a = a. a, then a-1 = —a.
a
- Equal vector : Two vectors a and b are equal if they have same magnitude, as well the same direction and written as a =
Obviously if a = b, then a = b.
- Negative of a vector : A vector with the same magnitude as that of a given vector a and direction opposite to that of a, is called the negative of a and denoted by –
Thus if AB = a, then – a = BA.
- Collinear vector : Two or more vectors with the same support are said to be
- Like and unlike vectors : Vectors having the same direction are known as like vectors while those having opposite direction are known as unlike vectors. These are collinear
- Coplanar vectors : All vectors are known to be coplanar when they are parallel to the same
- Co-initial vector : Vectors having the same initial point are known as co- initial
- Localised vectors and free vectors : A vector parallel to a given vector through a specified point as the initial point is known as a localised If the initial point of a vector is not specified, it is known to be a free vector.
- Position vector : Suppose O is the origin and let A be a point such that OA
= a, then the position vector of A is a.
Some Properties of vectors and Formulae :
- Triangle law of vector addition : AB + BC = AC. If the two sides of a tri- angle represent two vectors in magnitude and direction then their sum is represented by the third side taken in opposite order.
- Parallelogram law of vector addition : If ABCD is a parallelogram, then AB + AD = AC. If the two adjacent sides of a parallelogram can represent two vectors in magnitude as well as in direction, then their sum is represented by the diagonal through that
- Commutative property : x + y = y + x
- Associative property : (x + y) + z = x + y (y + z) x + 0 = x = 0 + x
x + (- x) = 0 = – x + x,
x + (- y) = x – y,
– (- x) = x
- (1) x and y are parallel if and only if x = my for some non-zero scalar m. 1
- x =—— or x = |x| x
|x|
- r, x, y are said to be coptanar if and only if r = a1x + b1y for some scalar a1 and b1.
AB = y – x
If r = x i + y j + z k, then |r| = Öx2 + y2 + z2
- Three points x, y, z are said to be collinear if and only if xy = mxz, for some non-zero scalar
- Three vectors a1x + b1y + c1z, a2x + b2y + c2z, a3x + b3y + c3z, where x, y, z are non-coplanar vectors, are said to be
a1 b1 c1 coplanar if and only if a2 b2 c2 0.
a3 b3 c3
- |x| – |y| £ |x + y| £ |x| + |y|
- |x – y| ³ |x| – |y|
- y = xy cos –, where 0 £ q £ p x.y = x. (Projection of y along x) Projection y.x
of y along = —–
|y|
- The cross product x C y tells two vectors perpendicular to both x and
(x C y)
- The unit vector perpendicular to both x and y is given by n = ——–
|x C y|
- x.y = 0 Þ x = 0 or y = 0
- Component of a vector r in the direction of a and perpendicular to a
r.a r.a
are —- a and r———- a respectively.
|a| |a|2
- If x and y are the non-zero vectors, then x.y = 0 Û x ^ y xy
cos Þ = x.y = —-
xy
- x.x = x2 = x2 = |x|2, |x| = Öx.x,
- |x.y| £ |x| |y|
- i.i = j.j = k.k = 1
i.j = j.j = j.k = k.j = k.i = i.k = 0
- If x = (x1, x2, x3) and y = (y1, y2, y3) e., if x = x1i + x2j + x3k and y = y1i + y2j + y3k, then
- y = x1y1 + x2yx + x3yc cos — x1y1 + x2y2 + x3y3
=
x 2 + x 2 + x 2 y 2 + y 2 + y 2
1 2 3 1 2 3
- x and y will be perpendicular if and only if x1y1 + x2y2 + x3y3 = 0
- a and b will be parallel if and only
x1 x2 x3 if = =
y1 y2 y3
- x C y = xy sin q
- |x C y| = xy sin q
- i C i = j C j = k C k = 1, i C j = k, j C k = i, k C i = j,
j C i = – k, k C j = – i, i C k = – j
- x C y = – y C x
- x C (y C z) ¹ (x C y) C z
- x C y = (x2y3 C x3y2) i + (x3y1 – x1y3) j + (x1y2 – x2y1) k
i j k x C y = x1 x2 x3
y1 y2 y3
- Area of the parallelogram ABCD = |AB CAD| or 1/2 |AC C BD|
- Area of the triangleABC = 1/2 |AB C AC|
- Unit vectors perpendicular to both
x C y
x and y are =——– –.
|x C y|
- If x =x1i + x2j + x3k, y = y1i + y2j + y3k and z = z1j + z2j + z3k, then
x1 x2 x3 (x C y).z = y1 y2 y3
z1 z2 z3
[x y z] = Volume of the parallelopiped whose coterminous edges are formed by x,
y, z.
[x y z] = [y z x] = [z x y]
but [x y z] = – [x z y] etc.
- Change of any two vectors in scalar triple product changes the sign of the
scalar triple product.
- If any three vectors x, y, z are equal, then [x y z] = 0
- The position of dot and cross in a scalar triple product can be Hence [i j k] = 1
- If l is a scalar, then [l x y z] = l[x y z] [x + d y t] = [x y z] + [t y z]
- x, y, z are coplanar if and only if [x y z] = 0
Vector triple product :
x C (y C z) = (x.z)y – (x.y) z,
(y C z) C x = (x.y)z – (x.z) y
The vector triple product is not commutative i.e., x C (y C z) ¹ (x C y) C z. Lagrange’s identity :
x.z x.t (x C y).(z C t) = y.z y.t
= (x.z) (y.t) – (x.t) (y.z)
Reciprocal system of vectors : If x, y, z are any three non-coplanar vectors so that [x y z] ¹ 0, then the three vectors x¢, y¢, z¢ defined by the equations
y C z z C x x C y
x¢ = ——- , y¢ = ——- , z¢ =——– , are called reciprocal vectors to the given
[xyz] [xyz] [xyz] Vectors x, y, z.
x.x¢ = y.y¢ = z.z¢ = 1
The scalar product of any vector of the other system which does not correspond
to it, is zero. i.e., x.y¢ = x.z¢ = y.x¢ = y.z¢ = z.x¢ = z.y¢ = 0 [xyz] [x¢y¢z¢] = 1
i¢ = i, j¢ = j, k¢ = k
if (x¢,y¢,z¢) is reciprocal system of (x, y, z) and r is any vector, then r = (r.x) x¢ + (r.y)y¢ + (r.z)z¢
r = (r.x¢) x + (r.y¢)y + (r.z¢) z
Applications of Vectors in Mechanics :
- Work done = (Force) C
- Moment of a force F about a point O = OP C F, where P is a point on the line of action of the force
- Moment of the couple (F.d) = d C
Product of two vectors : The product of two vectors is of two types (1) scalar product or dot product and (2) vector product or cross product.
Scalar (Dot Product) : If the product of two vectors gives a scalar quantity then it is known as scalar or dot product and is expressed as x.y = |x| |y| cos q
Where q is the angle between the two vectors. 0 £ q £ p
Using the components of the vectors along the cartesian axes the graduated product takes the form
x.y = x1y1 + x2y2 + x3y3
with x = x1i + x2j + x3k and y = y1i + y2j + y3k
Properties of Scalar Product : (1) Commutative Property x.y = y.k
- Distributive Property (y + z) = x.y + x.z
- Association with a scalar, if p is a sclar then p(x.y) = (px).y = x(p.y)
- Two vectors are perpendicular if and only if x.y = 0. Thus we have i.j = j.k = k.i = 0
Note : If two vectors are perpendicular to each other then the dot products is nil.
- If two vectors are parallel, their x.y = |x|.|y| and we have x = |x| also i2 = j2 = k2 = 1
Two Identities for Two Vectors x and y
(1) (x + y)2 = x2 + 2xy + y2 (2) x2 – y2 = (x + y)(x – y)
Vector Product or Cross Product : The vector product of two vectors x.y is the vector xy sin qu.
x.y = xy sinq, u 0 £ q £ p
Where q is the angle formed by the directions of x and y and u is a unit vector perpendicular to the plane of x and y so that x,y,u form a clockwise system.
The cross product between two vectors gives the vector area of a parallelogram whose adjacent sides represent the two vectors x and y.
Using the components of vectors along the cartesian axes, the vector product takes the form
i j k
x C y = x1 x2 x3 x C y y1 y2 y3
Properties of vector product : (1) If two vectors are parallel (i.e. q = 0 or. –) then x C y = 0 and we have i2 = j2 = k2 = 0,x C x = 0
- If two vectors are perpendicular
p
(i.e. q = —)
2
Then x C y = |x|.|y|n and |x C y| = |x||y|, we have i.j = k, k.j = i,k.i = j
- x C y = -y.x (anticommutative property)
- x C (y C z) = x C y + x C z (distributive property)
- Association with a scalar : If l is a scalar, then
l (x C y) = (lx) C y = x C (ly)
To find vector product of two vectors x = x1i + x2j + x3k and y = y1i + y2j + y3k We use
i j k
x C y = x1 x2 x3 y1 y2 y3
Also we have
(x C y) Sin2q = ——–
|x|2|y|2
Relation between Vector Product and Scalar
|x C y| = Öx2 .y2 – (x.y)2
Product of three vectors
- Scalar triple product : The product (y C z) or (x C y).z is known as scalar triiple product.
Mathematically
x.(y C z) = |x||y||z| sin q cos j where q is the angle between vector y and vector z and j is the angle between vector x and the resultant vector of product y C z.x.(y C z) is sometimes written as (x.yz)
———————————
y x z ——————————–
———————————-
If x = x1i + x2j + x3k y = y1i+ y2j + y3k and z = z1i + z2j + z3k
x1 x2 x3
then x.(y C z) = y1 y2 y3 z1 z2 z3
Properties of Scalar Triple Product
- The scalar triple product x.(y C z) gives the volume of the parallelopiped formed by the three vectors originating from a common point.
- x.(y C z) = y.(z C x) = z.(x C y) = (x C y).z = (y C z).x = (z C x).y
We can see that in a scalar triple product the signs of dot and cross can be interchanged.
- x.(y C z) = x.(z C y)
- Scalar triple product of three non-zero vectors is zero when the vectors are coplanar or we can say that three non-zero vectors will be If y.(x C z) = 0
x1 x2 x3
y1 y2 y3 = 0
z1 z2 z3
- The scalar triple product is
