1 CHAPTER 3 VECTORS NHAA/IMK/UNIMAP
2 INTRODUCTION Definition 3.1a 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
3 INTRODUCTION Definition 3.2a 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
4 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
5 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
6 INTRODUCTION Definition : Component FormIf 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
7 INTRODUCTION Definition : Magnitude / Lengththe magnitude of the vector is: NHAA/IMK/UNIMAP
8 Example 1 Find : component form andlength of the vector with initial point P(-3,4,1) and terminal point Q(-5,2,2) NHAA/IMK/UNIMAP
9 VECTOR ALGEBRA OPERATIONSDefinition : Vector Addition and Multiplication by a Scalar Let and be vectors with a scalar, k. ADDITION SCALAR MULTIPLICATION NHAA/IMK/UNIMAP
10 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. NHAA/IMK/UNIMAP
11 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
12 SUBTRACTION OF VECTORSThe subtraction of 2 vectors, u and v is defined by: If and then, NHAA/IMK/UNIMAP
13 SUBTRACTION OF VECTORSThe subtraction of 2 vectors, u and v is defined by: u v u+v -v u-v NHAA/IMK/UNIMAP
14 THE SUM OF A NUMBER OF VECTORSThe 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
15 SCALAR MULTIPLICATIONS OF VECTORSDefinition : 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
16 Example Let and . Find: NHAA/IMK/UNIMAP
17 PROPERTIES OF VECTOR OPERATIONSLet u, v, w be vectors and a,b be scalars: NHAA/IMK/UNIMAP
18 UNIT VECTORS DefinitionIf 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
19 DIRECTIONS OF ANGLES & DIRECTIONS OF COSINESz 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
20 Example Find the direction cosines and direction angles of: NHAA/IMK/UNIMAP
21 DOT PRODUCT Also known as inner product or scalar productThe result is a SCALAR If and then: NHAA/IMK/UNIMAP
22 Example If and : NHAA/IMK/UNIMAP
23 DOT PRODUCT Angle Between 2 VectorsIf the vectors lies on the same line or parallel to each other, then NHAA/IMK/UNIMAP
24 Example Find the angles between and NHAA/IMK/UNIMAP
25 Properties of Dot Productif u and v are orthogonal NHAA/IMK/UNIMAP
26 CROSS PRODUCT The result is a vector If and then: NHAA/IMK/UNIMAP
27 Example 3 Find the cross product between and NHAA/IMK/UNIMAP
28 CROSS PRODUCT Properties of Cross Product if u and v are parallelNHAA/IMK/UNIMAP