Chapter 2 - Vector Algebra

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1 A spatial vector, or simply vector, is a concept characterized by a magnitude and a direction, and which sums with other vectors according to the Parallelogram Law. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A to a terminal point B. This vector is commonly denoted by as shown in the fig. indicating that the arrow points from A to B. In this way, the arrow holds all the information of the vector quantity the magnitude is represented by the arrow s length and the direction by the direction of the arrow s head and body. This magnitude and direction are those necessary to carry a point from A to B. Note : Two vectors are said to be equal provided they have the same direction and magnitude. For example : from the fig. are equal. since they have the same direction and magnitude Both the aspects length and magnitude have to be taken into account for two vectors to be equal. Caution : i) Two vectors cannot be said to be equal, if they have only the same direction. ii) Two vectors cannot be said to be equal, if they have only the same magnitude. For example : are not equal as their magnitudes (lengths) are diferent even though their directions are equal For example : In rhombus ABCD (as shown in the fig) all the sides are equal in magnitude. But are not equal as their directions are different even though their lengths are equal. Note that in rhombus ABCD are equal. are equal since their magnitudes (lengths) and directions are equal. Parallelogram Law of Forces in Physics states that, if the forces 61 act along the sides of a Parallelogram ABCD as shown in the fig. then their resultant obtained by their sum acts along the diagonal AC. This is also known as Triangle law of Addition. i.e. Chapter 2 - Vector Algebra

2 Vectors have a variety of algebraic properties. Vectors may be scaled by stretching them out, or compressing them. They can be flipped around so as to point in the opposite direction. Two vectors sharing the same initial point can also be added or subtracted. Use of Vectors in Physics and Engineering Vectors are fundamental in the Physical Sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity metre / sec. 5 up could be represented by the vector (0,5). Another quantity represented by a vector is force, since it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement, acceleration, eletrical and magnetic fields, momentum, and angular momentum The importance of Vector Algebra Learning Vector Algebra is an important step that enhances the students ability in solving problems. The importance of Vector Algebra can be understood in the context of what has been learnt earlier : In Middle school or High School students are taught basic algebra and arithmetic, which will not be sufficient to solve most real-world problems. For example, a student may be asked to find the speed required to travel 33 miles in 60 minutes. For this problem, simple arithmetic is just enough. However if the student is asked to find the position of the vehicle after travelling a distance of 33 miles, arithmetic alone is not of much use. To overcome this difficulty the knowledge of Vector Algebra is important. At later stages, the students begin to realize that even algebra and arithmetic cannot solve problems that incorporate two-dimensional space, so they learn trigonometry and geometry. For example, if a student tries to find the amount of concrete needed to fill a cone-shaped hole, simple algebra alone will be of little help. However, geometry and trigonometry are very difficult to apply in many situations. Vector Algebra was invented in order to solve two-dimensional and threedimensional problems without the use of cumbersome geometry. Although it is possible to use ordinary trigonometry and geometry to solve most of the problems in Physics, students are likely to encounter, Vector Algebra has some significant advantages: 1. Vector Algebra is much easier to apply than geometry and requires knowledge of fewer rules. 2. The mechanics of Vector Algebra are straightforward, requiring less intuition and cleverness in finding a solution. (Geometry proofs in High School might have been difficult) 62

3 Important Points in Vector Algebra Important Points : 1. Dot Product or Scalar Product : The Dot Product of two Vectors that are inclined at an angle is a real number equal to ab cos denoted as and is read as 2. When (since cos 90 0 = 0 & cos = 0) 3. Let = a 1 b 1 + a 2 b 2 + a 3 b 3 The dot product of two vectors is obtained by multiplying the like co-efficients of both the vectors and then adding them 4. Let θ be the angle between. The formula for finding the angle between is 5. i) Projection of ii) Projection of 6. Distributive property is

4 9. i) ii) iii) iv) Cross Product or Vector Product Cross product or Vector product of two vectors that are inclined at angle is a vector quantity equal to denoted as and is read as cross where is the unit vector to & and has direction along ON Note : [Cross product is not commutative but dot product is commutative] In fact Note : i) When are Vectors (direction of may be same or opposite) (i.e). Note : i) If m is +ve the direction of are the same and if m is ve, the direction of are opposite. ii) vector. iii) i.e If the anticlockwise order is preserved, positive sign is given for the i.e If the clockwise order is preserved negative sign is supplied for the vector. 64

5 iv) Let be the two vectors. i.e co-eff. of in the 2 nd row and the co-eff. of in the 3 rd row v) Unit Vector to is vi) vii) Area of a triangle, given two vector sides ix) Formula for finding the angle between two vectors using vector product is, where θ is the angle between x) The three points A, B, C, whose position vectors are any one of the following conditions is satisfied i) ii) iii) [where l,m,n are any scalars] Physical application of dot product and cross product of vectors 1. The formula for work done by a force is Note : When two or more forces are given add the forces to get the resultant vector 2. The formula for Moment of Force (or) torque is given by 65 Scalar Triple Product

6 The three vectors are coplanar if any one of the following conditions is satisfied. 4. Four points A, B, C and D are coplanar if box product If the cyclic order is preserved then the value of the Scalar Triple Product does not change. 7. If the cyclic order is not preserved, then the sign of the Scalar Triple Product changes. Vector Triple Product i) linear combination of ii) 4. i) linear combination of 66

7 ii) Product of Four Vectors Straight lines Straight Lines and Planes 1. a) is the vector equation of the line through a given point A and parallel to b) is the cartesian equation of the line through the point A ( x 1 y 1 z 1 ) and parallel to the vector 2. a) is the vector equation of the line through the points A and B b) is the cartesian equation of the line through the points A( x 1 y 1 z 1 ) and B( x 2 y 2 z 2 ) 3. Angle between the lines 67 Note : The angle between the lines is the angle between their parallel vectors Planes 1. The vector equation of the plane passing through a given point and parallel to two

8 given vectors is The corresponding Cartesian equation is = 0 The corresponding vector equation in the non-parametric form is 2. The Vector equation of the plane passing through two given points parallel to a given vector is The corresponding Cartesian equation is The corresponding vector equation in the non-parametric form is 3. The Vector equation of the plane passing through three given non-collinear points is The corresponding Cartesian equation is The corresponding vector equation in the non-parametric form is 4. The Vector equation of the plane in the normal form is where is the unit vector to the plane and p is the length of the from the origin to the plane. The corresponding Cartesian equation is l x + my + nz = p Note : l, m, n are the direction cosines of the normal to the plane. 5. The Vector equation of the plane passing through a given point A and perpendicular (normal) to a given vector is 68 through which the straight line passes.

9 The corresponding Cartesian equation is a(x x 1 ) + b(y y 1 ) + c(z z 1 ) = 0 where a, b, c are direction ratios of the normal to the plane 6. Angle between the two planes is nothing but the angle between the normals to both the planes. Let If is the angle between the two planes then cos = where are the vectors normal to the two given planes. 7. Angle between the line and the plane is the complimentary angle of the line and the normal to the plane. Let If is the angle between the line and the plane then where is the vector parallel to the line and is the vector perpendicular to the plane 8. Distance of a point whose position vector is from the plane is the unit vector to the plane and p is the length of the from the origin to the plane. 9. Distance of a point A(x 1 y 1 z 1 ) to the plane ax + by + cz + d = 0 is 69 Skew lines Let L 1 and L 2 be two straight lines in space. The lines that are non-parallel and non-intersecting and lying in two different planes are called Skew lines. The shortest distance between two Skew lines is that line segment which is perpendicular to both the lines. i) Let be the two parallel lines parallel to and passing through the points A 1 and A 2 whose position vectors are The distance between the two parallel lines is given by the formula

10 are the position vectors of the points A 1 and A 2, through which the straight lines pass and both the lines. is the vector which is parallel to ii) Let be any skew lines. The first line passes through the point A 1 with position vector and parallel to. The second line passes through the point A 2 with position vector and parallel to. The shortest distance between these two skew lines is given by the formula Note : When the distance between two non-parallel lines (skew lines) in space is zero, then the two lines are coplanar (i.e the two lines lie on the same plane - i.e the two lines are intersecting lines) i.e If the two lines are intersecting lines i.e. they are coplanar. (or) If 70