CBSE Class 12 Commerce Mathematics Vectors
CBSE Class 12 Commerce Mathematics Vectors Introduction:-In our day to day life, we come across many queries such as – What is your height? How should a football player hit the ball to give a pass to another player of his team? Observe that a possible answer to the first query may beCBSE Class 12 Commerce Mathematics Vectors
1.6 meters, a quantity that involves only one value (magnitude) which is a real number. Such quantities are called scalars. However, an answer to the second query is a quantity (called force) which involves muscular strength (magnitude) and direction (in which another player is positioned). Such quantities are called vectors. In mathematics, physics and engineering, we frequently come across with both types of quantities, namely, scalar quantities such as length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance etc. and vector quantities like displacement, velocity, acceleration, force, weight, momentum, electric field intensity etc.
In this chapter, we will study some of the basic concepts about vectors, various operations on vectors, and their algebraic and geometric properties. These two type of properties, when considered together give a full realisation to the concept of vectors,
and lead to their vital applicability in various areas as mentioned above.
CBSE Class 12 Commerce Mathematics Vectors:-Some Basic Concepts
Let ‘l’ be any straight line in plane or three dimensional space. This line can be given two directions by means of arrowheads. A line with one of these directions prescribed is called a directed line
CBSE Class 12 Commerce Mathematics Vectors:-Direction Cosines
Consider the position vector OP(or ) ???? ?r of a point P(x, y, z) as in Fig 10.3. The angles α, β, γ made by the vector r? with the positive directions of x, y and z-axes respectively, are called its direction angles. The cosine values of these angles, i.e., cosα, cosβ and cos γ are called direction cosines of the vector r? , and usually denoted by l, m and n, respectively.
CBSE Class 12 Commerce Mathematics Vectors:-Types of Vectors
Zero Vector A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as 0 ? . Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors AA, BB
???? ???? represent the zero vector,
Unit Vector A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of a given vector a? is denoted byˆa .
Coinitial Vectors Two or more vectors having the same initial point are called coinitial vectors.
Collinear Vectors Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
Equal Vectors Two vectors a and b ? ? are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as a = b ? ? .
Negative of a Vector A vector whose magnitude is the same as that of a given vector(say, AB ???? ), but direction is opposite to that of it, is called negative of the given vector. For example, vector BA ???? is negative of the vector AB ???? , and written as BA = − AB ???? ???? .
CBSE Class 12 Commerce Mathematics Vectors:-Addition of Vectors
A vector AB ???? simply means the displacement from a point A to the point B. Now consider a situation that a girl moves from A to B and then from B to C (Fig 10.7). The net displacement made by the girl from point A to the point C, is given by the vector AC ???? and expressed as .
This is known as the triangle law of vector addition.
CBSE Class 12 Commerce Mathematics Vectors:-Multiplication of a Vector by a Scalar
Let a ? be a given vector and λ a scalar. Then the product of the vector a ? by the scalar λ, denoted as λa ? , is called the multiplication of vector a? by the scalar λ. Note that, λa ? is also a vector, collinear to the vector a? . The vector λa ? has the direction same (or opposite) to that of vector a ? according as the value of λ is positive (or negative). Also, the magnitude of vector λ a? is |λ| times the magnitude of the vector a ? , i.e.,
This form of any vector is called its component form. Here, x, y and z are called as the scalar components of r? , and xiˆ, yˆj and zkˆ are called the vector components of r? along the respective axes. Sometimes x, y and z are also termed as rectangular components.
CBSE Class 12 Commerce Mathematics Vectors:-Section formula
Let P and Q be two points represented by the position vectors OP and OQ ???? ???? , respectively, with respect to the origin O. Then the line segment joining the points P and Q may be divided by a third point, say R, in two ways – internally (Fig 10.16) and externally (Fig 10.17). Here, we intend to find the position vector OR ???? for the point R with respect to the origin O. We take the two cases one by one.
CBSE Class 12 Commerce Mathematics Vectors:-Product of Two Vectors
So far we have studied about addition and subtraction of vectors. An other algebraic operation which we intend to discuss regarding vectors is their product. We may recall that product of two numbers is a number, product of two matrices is again a matrix. But in case of functions, we may multiply them in two ways, namely, multiplication of two functions pointwise and composition of two functions. Similarly, multiplication of two vectors is also defined in two ways, namely, scalar (or dot) product where the result is a scalar, and vector (or cross) product where the result is a vector. Based upon these two types of products for vectors, they have found various applications in geometry, mechanics and engineering. In this section, we will discuss these two types of products.
CBSE Class 12 Commerce Mathematics Vectors:-Projection of a vector on a line
Suppose a vector AB ???? makes an angle θ with a given directed line l (say), in the anticlockwise direction (Fig 10.20). Then the projection of AB
???? on l is a vector p ? (say) with magnitude | AB| cosθ ???? , and the direction of p ? being the same (or opposite) to that of the line l, depending upon whether cos θ is positive or negative.
CBSE Class 12 Commerce Mathematics Vectors:-Vector (or cross) product of two vectors
In Section 10.2, we have discussed on the three dimensional right handed rectangular coordinate system. In this system, when the positive x-axis is rotated counterclockwise
Definition 3 The vector product of two nonzero vectors a and b ? ? , is denoted by a × b ? ? and defined as a × b ? ? = | a || b | sin θ nˆ ? ? ,
CBSE Class 12 Commerce Mathematics Vectors
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