# Vectors & Scalars - Welsh Government

Vectors and Scalars Copyright John OConnor St. Farnans PPS Prosperous For non-commercial purposes only.. Enjoy! Comments/suggestions please to the SLSS physics website forum @ http://physics.slss.ie/forum A scalar quantity is a quantity that has magnitude only and has no direction in space Examples of Scalar Quantities: Length Area Volume Time Mass A vector quantity is a quantity that has

both magnitude and a direction in space Examples of Vector Quantities: Displacement Velocity Acceleration Force Vector diagrams are shown using an arrow The length of the arrow represents its magnitude The direction of the arrow shows its direction The resultant is the sum or the combined effect of two vector quantities Vectors in the same direction:

6N 4N = 10 N = 10 m 6m 4m Vectors in opposite directions: 6 m s-1 10 m s-1 = 4 m s-1 6N

10 N = 4N When two vectors are joined tail to tail Complete the parallelogram The resultant is found by drawing the diagonal When two vectors are joined head to tail

Draw the resultant vector by completing the triangle 2004 HL Section B Q5 (a) Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? 5N Solution: Complete the parallelogram (rectangle) The diagonal of the parallelogram ac represents the resultant force The magnitude of the resultant is found using Pythagoras Theorem on the triangle a abc Magnitude ac 12 2 52 ac 13 N 12 Direction of ac : tan 5 12 tan 1 67

5 b 12 N d 13 N 12 Resultant displacement is 13 N 67 with the 5 N force 5 c Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. Solution:

Find the resultant of the two 5 N forces first (do right angles first) ac 52 52 50 7.07 N 5 tan 1 45 5 So, Resultant = 10 N 7.07 N = 2.93 N in the direction of the 10 N force c N 7. 07 5N 90 45 a N

N 2. 93 Now find the resultant of the 10 N and 7.07 N forces The 2 forces are in a straight line (45 + 135 = 180) and in opposite directions 10 5 d 135 5N 5 b

What is a scalar quantity? Give 2 examples What is a vector quantity? Give 2 examples How are vectors represented? What is the resultant of 2 vector quantities? What is the triangle law? What is the parallelogram law? When resolving a vector into components we are doing the opposite to finding the resultant We usually resolve a vector into components that are perpendicular to each other Here a vector v is resolved into an x component and a y component v y

x Here we see a table being pulled by a force of 50 N at a 30 angle to the horizontal When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N y=25 N N 50 30

x=43.3 N We can see that it would be more efficient to pull the table with a horizontal force of 50 N If a vector of magnitude v and makes an angle with the horizontal then the magnitude of the components are: v y=v Sin x = v Cos y y = v Sin

x=v x Cos Proof: x Cos v x vCos y Sin v y vSin 2002 HL Sample Paper Section B Q5 (a) A force of 15 N acts on a box as shown. What is the horizontal component of the force? Vertical Component y 15Sin60 12.99 N 15 N Horizontal Component x 15Cos 60 7.5 N

Vertical 12.99 N Component Solution: 60 Horizontal 7.5 N Componen t 2003 HL Section B Q6 A person in a wheelchair is moving up a ramp at constant speed. Their total weight is 900 N. The ramp makes an angle of 10 with the horizontal. Calculate the force required to keep the wheelchair moving at constant speed up the ramp. (You may ignore the effects of friction). Solution: If the wheelchair is moving at constant speed (no acceleration), then the force that moves it up the ramp must be the same as the component of its weight parallel to the ramp. Complete the parallelogram.

Component of weight 28 N 156. 10 parallel to ramp: 900Sin10 156.28 N Component of weight perpendicular to ramp: 900Cos10 886.33 N 80 10 900 N 886.33 N If a vector of magnitude v has two perpendicular components x and y, and v makes and angle with the x component then the magnitude of the components are: x= v Cos

y= v Sin y=v Sin v x=v Cos y

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