VECTOR Addition of vectors (parallelogram law of vector, law of triangle) Subtraction of vectors Resolving vectors Sine rule and cosine rule to find the resultant

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Scalar and Vector Scalar Quantities *Quantities which do not have direction but have magnitudes * Eg : mass, distance, speed, energy, temperature and pressure. Vector Quantities *Quantities which have both magnitude and direction * Eg : displacement, velocity, acceleration, force, momentum and field strength m E T p F B v p

Presentation of Vectors:

Presentation of Vectors On a diagram, each vector is represented by an arrow Arrow pointing in the direction of the vector Length of arrow is proportional to the magnitude of the vector Symbol or V Magnitude of the vector: V or

Addition and Subtraction of vector :

Addition of Vectors — Graphical Methods For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates. Addition and Subtraction of vector

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If the motion is in two dimensions, the situation is somewhat more complicated. Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.

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Adding the vectors in the opposite order gives the same result:

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Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

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The parallelogram method may also be used; here again the vectors must be tail-to-tip.

Parallelogram Rule:

Parallelogram Rule Draw the two vectors a and b Complete the paralellogram c is the resultant force. The direction is indicated by the angle α a b θ a b c θ α

Triangle Rule:

Triangle Rule Draw the 1 st vector Draw the 2 nd vector beginning at the head of the 1 st vector b b a θ 3. Complete the triangle by joining the tail of the 1 st vector to the head of the 2 nd vector for the resultant vector c . The direction of c is indicated by α . b a θ c α

Addition of Vectors : The law of vector addition:

Addition of Vectors : The law of vector addition Commutative rule Vector can be added in any order Expressed equation: A+B=C A B A+B=C B A

Addition of Vectors : The law of vector addition:

Addition of Vectors : The law of vector addition 2 ) Associative rule The resultant force does not depends on how the vectors are grouped Find (A+B)+C and A+(B+C) Expressed equation: (A+B)+C C A B A+B A+(B+C) C A B B+C

Addition of Vectors : The law of vector addition:

Addition of Vectors : The law of vector addition 3) Distributive rule The sum of two vector is multiplied by a positive scalar quantity, x then the resultant vector has the same direction but different in magnitude Find x ( A + B ) Expressed equation:

Example 1:

Example 1 Encik Rahim pushes a load with a force of 10 N while Mr. Lim pulls the same load with a force of 12 N as shown. What is the resultant force exerted by Encik Rahim and Mr. Lim on the load? F 1 =10 N F 2 =12 N

Example 2:

Example 2 A boat which can travel at a speed of 5 ms -1 in still water is sailing upstream in a river where the flow of water is 2 ms -1 . What is the resultant velocity of the boat?

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Subtraction of Vectors In order to subtract vectors, we define the negative of a vector , which has the same magnitude but points in the opposite direction . Then we add the negative vector.

Resolving Vectors:

Resolving Vectors 1) A vector B is shown as follow: x y 0 B 2) Draw a dotted line from the head of vector B to the x -axis. Then draw a straight line from the origin to the end point of the dotted line x y 0 B B x This is the x-component of vector B

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3) Draw a dotted line from the head of vector B to the y -axis. Then draw a straight line from the origin to the end point of the dotted line at the y-axis x y 0 B B x B y This is the y-component of vector B

Complete equation of the components:

Complete equation of the components x-component: y-component Magnitude of vector A

Example 3:

Example 3 Given that the magnitude and direction of vector B is 8 m and 53° respectively as shown in figure. Determine and draw its components. θ B x y Answer: B x =-6.39 m B y =-4.81 m

Example 4:

Example 4 Given that the magnitude and direction of vector C is 6.2 m and 28 respectively, as shown in figure. Determine and draw its components. θ C x y Answer: C x =5.47 m C y =-2.91 m

Vector Sign:

Vector Sign If x-comp is pointing to the right , then it is + ve If x-comp is pointing to the left , then it is – ve If y-comp is pointing upward , then it is + ve If x-comp is pointing downward , then it is – ve A x =+ ve B x =― ve B y =― ve A y =+ ve A B x y

Exercise:

Exercise Determine the resultant vector. Given that F 1 =10 N, F 2 =5 N, θ 1 =25° and θ 2 =30°. F 1 θ 1 θ 2 F 2 y x Answer: F x =4.733 N F y =1.726 N F =5.038 N θ=20.04°

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2) Given that F 1 =3 N, F 2 =6 N, F 3 =5 N, θ 1 =0°, θ 2 =15° and θ 3 =45°. Find the magnitude and direction of their sum. 3) Three vectors are specified as: A is 5 m at 45° north of east, B is 7 m at 60° east of south, and C is 4 m at 30° west to south. Find the magnitude and direction of their sum. 4) Given vector F 1 and F 2 are in the xy -plane. F 1 is 50 N at 45° and F 2 is 100 N at 120° counterclockwise. Determine the magnitude and direction of the resultant F x =12.34 N, F y =5.09 N F =13.35 N, θ=22.42° 6.45 m at 342.3° 122.84 N, 96.85°

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