GROUP’S MEMBERSName Matric No.Ridwan bin shamsudin D20101037472Mohd. Hafiz bin Salleh D20101037433Muhammad Shamim Bin D20101037460ZulkefliJasman bin Ronie D20101037474Hairieyl Azieyman Bin Azmi D20101037426Mustaqim Bin Musa D20101037402
INTRODUCTION . . Vectoris a variable quantity that can be resolved into components. Vector also is a straight line segment whose length is magnitude and whose orientation in space is direction. Hurmm . . .
SCALAR VECTOR VECTOR PRODUCT A scalar quantity has magnitude only A vector quantity has with an appropriate both magnitude and unit of direction. measurement. Examples of vector Example of scalar quantities are quantities are length, speed, time, displacement, temperatue, mass velocity, acceleration and power. and force.
The most commonly used example of vectors in everyday life is velocity. Vectors also mainly used in physics and engineering to represent directed quantities. Vectors play an important role in physics about of a moving object and forces acting on it are all described by vectors.
Since several important physical quantities are vectors, it is useful to agree on a way for representing them and adding them together. In the example involving displacement, we used a scale diagram in which displacements were represented by arrows which were proportionately scaled and orientated correctly with respect to our axes (i.e., the points of the compass).
Thisrepresentation can be used for all vector quantities provided the following rules are followed: 1.The reference direction is indicated. 2.The scale is indicated. 3.The vectors are represented as arrows with a length proportional to their magnitude and are correctly orientated with respect to the reference direction. 4.The direction of the vector is indicated by an arrowhead. 5.The arrows should be labelled to show which vectors they represent.
For example, the diagram below shows two vectors A and B, where A has a magnitude of 3 units in a direction parallel to the reference direction and B has a magnitude of 2 units and a direction 60° clockwise to the reference direction: I see ~
The length of a vector is called the magnitude or modulus of the vector. A vector whose modulus is unity is called a unit vector which has magnitude. The unit vector in the direction is called The unit vectors parallel to
The magnitude of vector a is written as |a|. The magnitude of vector AB is written as |AB|. 𝑥 𝑥2 + 𝑦2 𝑦 If a = then the magnitude |a|= *using pythagorean theorem.
EXAMPLES :1. Find the magnitude of the vector 2Solution :
A vector having the same magnitude but opposite direction to a vector A, is -A.If v is a vector, then -v is a vector pointing in the opposite direction.If v is represented by (a, b, c)T then -v is represented by (-a, -b, -c)T.
Example : Write down the negetive of solution : 3 = − −2 −3 = 2
• Is a vector with zero magnitude and no direction• |0|= 0
EXAMPLE : Determine whether w-y-x+z is a zero vector.SolutionFrom the diagram,w-y-x+z = OSince it does not has magnitude,thus it is a zerovector
2 vectors u and v are equal if their corresponding components are equal For example, if u=ai +bj and v=ci + dj then u = v a=c and b=d Or in another word we can say it is equal if the vectors have same magnitude and same direction
Vectors are parallel if they have the same direction Both components of one vector must be in the same ratio to the corresponding components of the parallel vector.(i) v1 kv2 , k any scalar (ii) v1 .v2 v1 v2 or v1 .v2 v1 v2 v x v 0 (iii) 1 2
EXERCISEExerciseGiven 2i-3j and 8i+yj are parallel vector. Find the value of y.SolutionSince they are parallel vectorsLet 8i+yj=k(2i-3j),k is any scalar 8i+yj=2ki-3kj8=2k y=-3kk=4 =-3(4) =-12
Algebraic properties of the scalar product for any vector a, b and c and m is a constant
1) a . a = a 22) a . b = b . a3) a . (b + c) = a . b + a . c4) (a b )c) (a b c) a b c 5) m (a . b) = (ma) . b = (a . b)m 6) a . b = a b if and only if a parallel to b a . b = – a b if and only if a and b in opposite direction7) a . b = 0 if and only if a is perpendicular to b8) .
Example: Evaluate a) (2 i j ) (3 i 4 k ) ~ ~ ~ ~ b) (3 i 2 k ) (i 2 j 7 k ) ~ ~ ~ ~ ~
Definition of VectorMultiplication In Vector Multiplication, a vector is multiplied by one or more vectors or by a scalar quantity.
More about VectorMultiplication There are three different types of multiplication: dot product, cross product, and multiplication of vector by a scalar. The dot product of two vectors u and v is given as u · v = uv cos θ where θ is the angle between the vectors u and v. The cross product of two vectors u and v is given as u × v = uv sin θ where θ is the angle between the vectors u and v. When a vector is multiplied by a scalar, only the magnitude of the vector is changed, but the direction remains the same.
Examples of VectorMultiplication If the vector is multiplied by a scalar then =. If u = 2i + 6j and v = 3i - 4j are two vectors and angle between them is 60°, then to find the dot product of the vectors, we first find their magnitude. Magnitude of vector Magnitude of vector The dot product of the vectors u, v is u · v = uv cos θ = (2 ) (5) cos 60° = (2 ) (5) × =5
If u = 5i + 12j and v = 3i + 6j are two vectors and angle between them is 60°, then to find the cross product of the vectors, we first find their magnitude. Magnitude of vector Magnitude of vector The cross product of the vectors u, v is u × v = uv sin θ = (3 ) (13) sin 60° = 39 (2) = 78
Solved Example on Vector Multiplication Which of the following is the dot product of the vectors u = 6i + 8j and v = 7i - 9j? Choices: A. 114 B. - 30 C. - 2 D. 110 Correct Answer: B Solution: Step 1: u = 6i + 8j, v = 7i - 9j are the two vectors. Step 2: Dot product of the two vectors u, v = u · v = u1v1 + u2v2 Step 3: = (6i + 8j) · (7i - 9j) Step 4: = (6) (7) + (8) (- 9) [Use the definition of the dot product of two vectors.] Step 5: = - 30 [Simplify.]
Definition of Addition ofVectors Adding two or more vectors to form a single resultant vector is known as Addition of Vectors.
More about Addition ofVectors If two vectors have the same direction, then the sum of these two vectors is equal to the sum of their magnitudes, in the same direction. If the two vectors are in opposite directions, then the resultant of the vectors is the difference of the magnitude of the two vectors and is in the direction of the greater vector.
Examples of Addition ofVectors To find the sum of the vectors of and , they are placed tail to tail to form two adjacent sides of a parallelogram and the diagonal gives the sum of the vectors and . This is also called as ‘parallelogram rule of vector addition’.
If the vector is represented in Cartesian coordinate, then the sum of the vectors is found by adding the vector components. The sum of the vectors u = <- 3, 4> and v = <4, 6> is u + v = <- 3 + 4, 4 + 6> = <1, 10>
Definition Of Subtraction OfVectors subtracting two or more vectors to form a single resultant vector is known as subtraction of vectors.
example f the vector is represented in Cartesian coordinate, then the subtraction of the vectors is found by subtracting the vector components. The sum of the vectors u = <- 3, 4> and v = <4, 6> is u - v = <- 3 - 4, 4 - 6> = <-7, -2>
The angle between 2 lines The two lines have the equations r = a + tb and r = c + sd. The angle between the lines is found by working out the dot product of b and d. We have b.d = |b||d| cos A.
Example Find the acute angle between the lines L : r i 2 j t (2i j 2k ) 1 L : r 2i j k s(3i 6 j 2k ) 2Direction Vector of L1, b1 = 2i –j + 2kDirection Vector of L2, b2 = 3i -6j + 2kIf θ is the angle between the lines, (2i j 2k ).( 3i 6 j 2k )Cos θ = 2i j 2k 3i 6 j 2k