CHAPTER 3 VECTORS NHAA/IMK/UNIMAP

INTRODUCTION Definition 3.1 a VECTOR is a mathematical quantity that has both MAGNITUDE AND DIRECTION VECTOR: represented by arrow where the direction of arrow indicates the DIRECTION of the vector & the length of arrow indicates the MAGNITUDE of the vector. Eg: displacement, velocity, acceleration, force, ect NHAA/IMK/UNIMAP

INTRODUCTION Definition 3.2 a SCALAR is a mathematical quantity that has MAGNITUDE only Scalar: represented by a single letter such as, k. Eg: temperature, mass, length area, ect NHAA/IMK/UNIMAP

INTRODUCTION Definition 3.3 A vector in the plane is a directed line segment that has initial point A and terminal point B, denoted by, ; its length is denoted by . Initial Point, A Terminal Point, B Length: NHAA/IMK/UNIMAP

INTRODUCTION Definition 3.5 Two vectors, and are said to be EQUAL if and only if they have the same MAGNITUDE AND DIRECTION. NHAA/IMK/UNIMAP

INTRODUCTION Definition : Component Form If v is a 3-D vector equal to the vector with initial point at the origin and the terminal point , then the component form of v is defined by: z y x v1 v3 v2 O NHAA/IMK/UNIMAP

INTRODUCTION Definition : Magnitude / Length the magnitude of the vector is: NHAA/IMK/UNIMAP

Example 1 Find : component form and length of the vector with initial point P(-3,4,1) and terminal point Q(-5,2,2) Answer a) b) NHAA/IMK/UNIMAP

VECTOR ALGEBRA OPERATIONS Definition : Vector Addition and Multiplication by a Scalar Let and be vectors with a scalar, k. ADDITION SCALAR MULTIPLICATION NHAA/IMK/UNIMAP

Initial point of v/ Terminal point of u ADDITION OF VECTORS The Triangle Law 2 vectors u and v represented by the line segment can be added by joining the initial point of vector v to the terminal point of u. Initial point of v/ Terminal point of u Terminal point of u Initial point of u NHAA/IMK/UNIMAP

ADDITION OF VECTORS The Parallelogram Law The sum, called the resultant vector is the diagonal of the parallelogram. u v u+v NHAA/IMK/UNIMAP

SUBTRACTION OF VECTORS The subtraction of 2 vectors, u and v is defined by: If and then, NHAA/IMK/UNIMAP

SUBTRACTION OF VECTORS The subtraction of 2 vectors, u and v is defined by: u v u+v -v u-v NHAA/IMK/UNIMAP

THE SUM OF A NUMBER OF VECTORS The sum of all vectors is given by the single vector joining the initial of the 1st vector to the terminal of the last vector. a b d c e NHAA/IMK/UNIMAP

SCALAR MULTIPLICATIONS OF VECTORS Definition : Let k be a scalar and u represent a vector, the scalar multiplication ku is: A vector whose length |k| time of the length u and A vector whose direction is: The same as u if k>0 and The opposite direction from u if k<0 NHAA/IMK/UNIMAP

Example Let and . Find: Answer a) b) c) NHAA/IMK/UNIMAP

PARALLEL VECTORS Definition : If two vectors, u and v have the same direction, whether their magnitudes are same or not, they are said to be parallel. To be parallel vectors, one should a scalar multiple of another. NHAA/IMK/UNIMAP

PROPERTIES OF VECTOR OPERATIONS Let u, v, w be vectors and a,b be scalars: NHAA/IMK/UNIMAP

UNIT VECTORS Definition If u is a vector, then the unit vector in the direction of u is defined as: A vector which have length equal to 1 is called a unit vector. NHAA/IMK/UNIMAP

DIRECTIONS OF ANGLES & DIRECTIONS OF COSINES z x y - Are the angles that the vector OP makes with positive axis - Known as the direction angles of vector OP P DIRECTION OF COSINES O NHAA/IMK/UNIMAP

Example Find the direction cosines and direction angles of: Answer (i) (ii) NHAA/IMK/UNIMAP

DOT PRODUCT Also known as inner product or scalar product The result is a SCALAR If and then: NHAA/IMK/UNIMAP

Example If and Answer 9 -18 NHAA/IMK/UNIMAP

DOT PRODUCT Angle Between 2 Vectors If the vectors lies on the same line or parallel to each other, then NHAA/IMK/UNIMAP

PERPENDICULAR (ORTHOGONAL) VECTORS Two vectors are perpendicular or orthogonal if the angle between them is Definition Vectors u and v are perpendicular (orthogonal) if and only if u.v =0. NHAA/IMK/UNIMAP

Example Find the angles between and Answer (i) (ii) NHAA/IMK/UNIMAP

Properties of Dot Product if u and v are orthogonal NHAA/IMK/UNIMAP

CROSS PRODUCT The result is a vector If and then: NHAA/IMK/UNIMAP

CROSS PRODUCT The vector is orthogonal to both u and v also known as the normal vector, n. NHAA/IMK/UNIMAP

Example 3 Find the cross product between and Answer (i) (ii) NHAA/IMK/UNIMAP

CROSS PRODUCT Properties of Cross Product if u and v are parallel NHAA/IMK/UNIMAP

VECTORS APPLICATIONS NHAA/IMK/UNIMAP

Line L is the set of all points P(x,y,z) for which parallel to : APPLICATIONS Lines & Line Segment in Space Parametric Equations z y x P0(x0,y0,z0) L P(x,y,z) v Line L is the set of all points P(x,y,z) for which parallel to : NHAA/IMK/UNIMAP

APPLICATIONS Therefore the parametric equation for L : Cartesian equation: Parametric Equation NHAA/IMK/UNIMAP

Example 1 Find parametric and Cartesian equation for the line passes through Q(-2,0,4) and parallel to Answer Parametric Equation Or Cartesian Equation NHAA/IMK/UNIMAP

Example 2 Find the parametric equation for the line passes through P(-3,2,-3) and Q(1,-1,4) Answer At point P At point Q NHAA/IMK/UNIMAP

APPLICATIONS Distance from point S to line L L v L From the properties of Cross Product Formula of Distance from point S to L NHAA/IMK/UNIMAP

Example 3 Find the distance from the point S(1,1,5) to the line Solution NHAA/IMK/UNIMAP

APPLICATIONS Equation of Planes Vector is on the plane M and vector which is perpendicular to M known as normal vector, n Equation of Plane is defined by: From the properties of Dot Product NHAA/IMK/UNIMAP

APPLICATIONS Equation of Planes Normal vector n : NHAA/IMK/UNIMAP

APPLICATIONS Let and EQUATION OF PLANE NHAA/IMK/UNIMAP

Example 4 Find an equation of plane through P0(-3,0,7) perpendicular to Answer NHAA/IMK/UNIMAP

Example 7: Find equation of plane through 3 points: Solution Find the normal vector, n: Equation of plane: (use any point A,B or C) NHAA/IMK/UNIMAP

APPLICATIONS Parallel Planes 2 planes parallel if and only if their normal planes are parallel NHAA/IMK/UNIMAP

APPLICATIONS Lines of Intersection 2 planes that are not parallel, intersect in a line. Normal vector for plane M and N Line of intersection NHAA/IMK/UNIMAP

APPLICATIONS Lines of Intersection From the properties of cross product, the result between vector product is a vector. Therefore, whenever two vectors crossing with each other, new vector will be produced. NHAA/IMK/UNIMAP

Example 8: Find a vector parallel to the line of intersection of the planes Answer NHAA/IMK/UNIMAP

Example 9: Find the parametric equation for the line in which the planes intersect. Solution Find v NHAA/IMK/UNIMAP

Find the intersection point: If x =0, Therefore, the point of intersection is The parametric equation for and point is: NHAA/IMK/UNIMAP

Example 9: Find the point where the line Intersects the plane Solution Find t Therefore, point of intersection is NHAA/IMK/UNIMAP

APPLICATIONS Distance from a Point to the Plane P n D P0 NHAA/IMK/UNIMAP

APPLICATIONS Equation of plane NHAA/IMK/UNIMAP

Example 10: Find the distance from S(1,1,3) to the plane Answer NHAA/IMK/UNIMAP