Chapter 3 VECTORS

OUTLINE OF CHAPTER 3: VECTORS Introduction Addition of Vectors Subtraction of Vectors Scalar Multiplication of Vectors PARALLEL VECTORS Components of Vectors IN TERMS OF UNIT VECTORS POSITION VECTORS Components of Vectors Magnitude of Vectors UNIT VECTORS OPERATIONS OF VECTORS BY COMPONENT Product of 2 Vectors Application of Scalar/Dot Product & Cross Product

VECTORS SCALARS 3.1 introduction Has magnitude only Has magnitude (represent by length of arrow) Direction (direction of arrow either to the right, left, etc) E.g: displacement, velocity, acceleration, force. - move the brick 5m to the right, SCALARS Has magnitude only No direction E.g: mass, temperature, length, area. - move the brick 5m

3.2 vector representation Use an arrow connecting an initial point A to terminal point B Written as a, w, or u. Magnitude of B (terminal point) B (initial point) A (initial point) A (terminal point)

3.3 vector negative The vector −a would represent the vector in opposite direction, but has the same magnitude as vector a. Geometrically, if B B A A

3.4 two equal vectors Two vectors a and b are said to be equal if and only if they have the same magnitude and direction.

3.5 addition of vectors (i) The Triangle Law Any 2 vectors can be added by joining the initial point of b to the terminal point of a. Example 3.1:

Exercise

(ii) The Parallelogram Law If 2 vector quantities are represented by 2 adjacent sides of a parallelogram, then the diagonal of parallel will be equal to the summation of these 2 vectors. Example 3.2: *The parallelogram law is affected by the triangle law.

(iii) The Sum of a Number of Vectors The sum of vectors is given by the single vector joining the start of the first vector to the end of the last vector. Example 3.3: *Note: The addition of vector is satisfied the commutative (a+b = b+a)

3.6 subtraction of vectors Using the parallelogram law of addition, a-b = a+(-b) Example 3.4:

3.7 scalar multiplication of vectors Let k be a scalar and a represent a vector, the scalar multiplication ka is: A vector whose length is |k| time that of the length a Example 3.5:

3.8 parallel vectors Two nonzero vectors a and b are parallel (a||b) , if one is a scalar multiple of the other. If a= αb with α>0 , then a and b have the same direction. If α<0, then a and b have opposite direction. E.g:

3.9 component of vectors (unit vectors) If we place the initial point of the vector u at the origin then the terminal point of u has coordinate <x, y, z>. These coordinate is called the components of u and we write u = xi + yj + zk where i, j and k are unit vectors in the x, y and z -axes, respectively: Unit vector (i, j, k ) i = <1,0,0> be a unit vector in the OX direction (x-axis) j = <0,1,0> be a unit vector in the OY direction (y-axis) k = <0,0,1> be a unit vector in the OZ direction (z-axis)

Vector in 2D (R2) Example 3.6:

3.10 POSITION vectors Vector in 3D (R3) Example 3.7: The position vector of the point with coordinates P(x, y, z) is where r represent the vector from the origin to point P in R3 .

3.11 component of vectors If is a vector with initial point P(a, b, c) and terminal point Q(x, y, z), then Example 3.8: Exercise 3.1:

3.12 magnitude (length) of vectors The magnitude or length of any vector a =<x, y, z> is Example 3.9: Exercise 3.2:

3.12 magnitude (length) of vectors The magnitude or length of any vector from point P(a, b, c) to point Q(x, y, z) is defined as Example 3.10: Exercise 3.3:

3.13 unit vectors If u is a vector, then the unit vector in the direction of u is defined as Example 3.11: Exercise 3.4:

3.15 operations of vectors by components If a=<a1, a2, a3>, b=<b1, b2, b3>, and k is a scalar, then Example 3.13: Exercise 3.6:

Properties of a vector If a, b and c are vectors, k and t are scalars,

3.16 product of two vectors The scalar/dot product of two vectors a=<a1, a2, a3> and b=<b1, b2, b3>, denoted a . b, is Example 3.14: Exercise 3.7:

If θ is the angle between the vectors a and b , then Theorem 1: If θ is the angle between the vectors a and b , then Note: a.b are orthogonal (perpendicular) if and only if a.b=0 Example 3.15: Exercise 3.8:

Theorem 1: (Angle between two vectors) If θ is the angle between the non-null vectors a and b , then θ is an acute angle if and only if a.b>0 θ is an obtuse angle if and only if a.b<0 θ =90O is a right angle if and only if a.b=0 Example 3.15:

Properties of the Scalar/Dot Product: If a and b are vectors, k is scalars, then

Exercise 1:

Exercise 2:

The cross product of two vectors a=<a1, a2, a3> and b=<b1, b2, b3>, denoted a x b by the determinant of the matrix Note: The result of vector product is a vector Example 3.16: Exercise 3.9:

If θ is the angle between the vectors a and b , then Theorem 2: If θ is the angle between the vectors a and b , then Note: axb=0 if and only if a//b (a and b are parallel) Example 3.17:

Exercise 3:

Properties of the Cross Product: If a, b and c are vectors, k is scalars, then

3.17 APPLICATIONS OF VECTORS Projections Equation of Planes Parametric Equations of Line Problems Involving Planes

Projections Consider the diagram below: Let θ Q S a The length of the shadow The scalar projection of vector b onto a (also called the component of b along a) is the length/magnitude of the straight line PS or which is

Consider the diagram below: Let P R b θ Q S a Shadow of b onto a The vector projection of vector b onto a is a vector in the direction of a with b magnitude equal to the length of the straight line PS or

Example 3.17: Exercise 3.10:

Equations of Planes Calculate the equation of the planes are defined as follows: i) with normal vector, through the point (1,0,-1). ii) through the origin and parallel to and . iii) through the three points (1,2,0), (3,1,1) and (2,0,0). iv) parallel to the plane in (a) and passing through the point (1,2,3).

Equations of Planes i) with normal vector, through the point (1,0,-1). P(x0,y0,z0) Q(x,y,z) n Suppose that is plane with the vector and a normal vector to the plane n. Let point P(x0,y0,z0) be an arbitrary point on and Q(x,y,z) be any point on plane . normal vector, n is orthogonal .

i) with normal vector, through the point (1,0,-1). Q(x,y,z) Then,

Exercise 4:

Example 3.20: Exercise 3.13:

Equations of Planes ii) through the origin and parallel to and . n u v

Equations of Planes iii) through the three points A(1,2,0), B(3,1,1) and C(2,0,0). A(1,2,0) B(3,1,1) C(2,0,0) Since A, B and C are coplanar, then the vector and are also coplanar. Thus will give a normal vector to the plane.

Equations of Planes iii) through the three points A(1,2,0), B(3,1,1) and C(2,0,0). A(1,2,0) B(3,1,1) C(2,0,0)

iv) parallel to the plane in (i) and passing through the point (1,2,3). Parallel to the plane from (i), x+2y+3z+2=0 and pass through the point (1,2,3)

Exercise 5:

Intersection of the plane Given two intersecting planes with angle θ between them. Let n1 and n2 be normal vector to these planes. Then, Thus two planes are Perpendicular if Parallel if where k is scalar.

Intersection of the plane

Exercise 6:

Parametric Equations of a Line Suppose L is a straight line that passes through the point P(x0,y0,z0) and is parallel to the vector as diagram below: O(0,0,0) v P(x0,y0,z0) Q(x,y,z) x y z O P Q Then another point Q(x,y,z) lies on the line L if and only if the vector v and are parallel:

Parametric Equations of a Line Q(x,y,z) P(x0,y0,z0)

Example 3.21: Exercise 3.14:

Exercise 7:

Intersection of two lines “intersecting” “parallel” “skew”=that the lines are neither parallel nor intersecting

Example: Exercise 7:

Problems involving planes We shall discuss about the shortest distance from a point to a plane. We consider 3 cases as follows: i) The distance from a point to a plane. ii) The distance between two parallel planes.. iii) The distance between two skewed lined on the plane. P0(x0,y0,z0) n D P1(x,y,z) P D P D (i) (ii) (iii)

i) The distance from a point to a plane. P0(x0,y0,z0) D P1(x,y,z) projection of b onto normal vector, n. Thus, Since P0(x0,y0,z0) lies on the plane, then this point satisfy the equation of plane, that is Thus, the formula D can be written as, Let P0(x0,y0,z0) be a point on the plane and b be the vector corresponding to Then, From the figure above, the shortest distance, D from P1 to the plane is equal to the absolute value of scalar

ii) The distance between two planes The shortest distance between two parallel planes ax+by+cz+d1=0 and ax+by+cz+d2=0 is given by

Example 3.22: Exercise 3.15:

ii) The distance between the skewed lines Example: Find the distance between the skewed lines as below, P1 D P2

Exercise 8: