1.4 Scalars and Vectors Objectives: • • • Understand the difference between scalar and vector quantities and give examples of scalar and vector quantities included in the syllabus Add and subtract coplanar vectors Represent a vector as two perpendicular components Scalar Vector Definition Physical quantities that have magnitude only. Physical Quantities that have both magnitude and direction. Examples Distance Speed Mass Energy Time Work Power Displacement Velocity Acceleration Force Momentum Weight Representation A scalar is represented by its magnitude (numeric value) and unit only. A vector is represented by its direction such as North, west, east, south or degrees or with arrows along with its magnitude and unit. Representing Vectors: Length of the arrow, drawn to scale represent magnitude, while where the arrow is pointing with respect to the reference represent direction. Page 1 of 6 Madiha Shakil Parallel Vectors: Vector Addition (Head to tail rule) – Graphical Method: (Head to tail rule) • Choose a suitable scale. Your diagrams should be reasonably large (covering the area given). • Draw a line to represent the first vector. • North at the top of the page. You will connect the head of one vector to the tail of the another. Vectors can be moved parallel to themselves without changing the resultant. The Red Arrow represents the resultant of the two vectors. Vector Subtraction – Graphical Method: Page 2 of 6 Madiha Shakil Coplanar Vectors: When 3 or more vectors need to be added, the same principles apply, provided the vectors are all on the same plane i.e., coplanar To subtract 2 vectors, reverse the direction i.e., change the sign of the vector to be subtracted, and add. ⃗𝑨 + ⃗𝑩 ⃗ + ⃗𝑪 = ⃗𝑹 ⃗ Page 3 of 6 Madiha Shakil Mathematical Methods: If the two vectors are at right angle to each other, we can use Pythagoras theorem to solve it To find magnitude 𝑪𝟐 = 𝑨𝟐 + 𝑩𝟐 To find direction tan 𝜃 = 𝒐𝒑𝒑 𝒂𝒅𝒋 Resolution of Vectors: We know when two vectors are added together, it can be represented by a single resultant vector that has the total/same effect of the two vectors. Now we are going to be resolving a single vector into two components. “if two perpendicular vectors have a resultant vector, that resultant vector can also be resolved into its two perpendicular vectors” We resolve a vector into two components Component along X-axis called x-component - See the green arrows to know how to represent them. Component along Y-axis called Y-component Which are perpendicular to each other. These components are called rectangular components of vector. Page 4 of 6 Madiha Shakil Addition of vector by rectangular components method: Consider two vectors acting in different directions. 1. Resolve both vectors into two rectangular components. 𝒗𝟏𝑿 = 𝒗𝟏 𝐜𝐨𝐬 𝜽𝟏 And 𝒗𝟏𝒀 = 𝒗𝟏 𝐬𝐢𝐧 𝜽𝟏 2. Resolve the second vector into its components. 𝒗𝟐𝑿 = 𝒗𝟐 𝐜𝐨𝐬 𝜽𝟐 And 𝒗𝟐𝒀 = 𝒗𝟐 𝐬𝐢𝐧 𝜽𝟐 Consider two vectors acting in different directions. 3. Find the magnitude of the x-component and ycomponent of the resultant. 𝑹𝒙 = 𝒗𝟏𝑿 + 𝒗𝟐𝑿 And 𝑹𝒚 = 𝒗𝟏𝒀 + 𝒗𝟐𝒀 4. Find the total magnitude of the resultant vector. 𝑹 = √𝑹𝟐𝑿 + 𝑹𝟐𝒀 5. 6. Find the angle of the resultant vector. 𝑹𝒀 𝜽 = 𝐭𝐚𝐧−𝟏 𝑹𝑿 Adding coplanar vectors by determining perpendicular components: Page 5 of 6 Madiha Shakil The total force on the object in the xcomponents of 𝑭𝟏 and 𝑭𝟐 𝑭𝑿 = 𝑭𝟏 𝐜𝐨𝐬 𝜽𝟏 + 𝑭𝟐 𝒄𝒐𝒔 𝜽𝟐 The total force on the object in the ycomponents of 𝑭𝟏 and 𝑭𝟐 𝑭𝒚 = 𝑭𝟏 𝐬𝐢𝐧 𝜽𝟏 + (−𝑭𝟐 ) 𝐬𝐢𝐧 𝜽𝟐 Negative sign to indicate opposite directions. Condition for Equilibrium: Coplanar vectors can be represented by vector triangles. In equilibrium, these are closed vector triangles. The vectors, when joined together, form a closed path If three forces acting on an object are in equilibrium; they form a closed triangle. Summary: Page 6 of 6 Madiha Shakil