Resultant of Perpendicular Vectors Revision Notes for Grade 11 NSC Matric Physical Sciences

Resultant of Perpendicular Vectors

Introduction to vectors in two dimensions

A vector is a physical quantity that has both magnitude (size) and direction. In Grade 10, you studied vectors in one dimension, but now we extend this concept to work with vectors in two dimensions using the Cartesian coordinate system.

When we represent vectors visually, we use arrows where the length indicates the magnitude and the arrowhead shows the direction. The starting point is called the tail and the endpoint is called the head.

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When writing a vector symbol, we place an arrow above the letter (e.g., $\vec{F}$ ) to show it represents a vector quantity with both magnitude and direction. This distinguishes it from scalar quantities which only have magnitude.

Vectors on the Cartesian plane

The Cartesian coordinate system consists of two perpendicular axes: the x-axis (horizontal) and y-axis (vertical). This system allows us to represent vectors in two dimensions rather than just along a single line.

Vector equality and positioning

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Key Concept: Vector Equality

Vectors are equal if they have the same magnitude and direction, regardless of where they are positioned on the coordinate plane. This means multiple arrows of the same length pointing in the same direction represent the same vector.

The position of a vector's starting point does not affect the physical quantity it represents, as long as the magnitude and direction remain unchanged.

Methods for finding resultants

When multiple vectors act together, we can find their combined effect using a single resultant vector. There are two main graphical methods for determining resultants.

Head-to-tail method

This method involves connecting vectors end-to-end in sequence using the head-to-tail method:

Draw the first vector from the origin
Place the tail of the second vector at the head of the first vector
Continue this process for all vectors
The resultant is drawn from the tail of the first vector to the head of the last vector

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The order in which you draw the vectors does not matter - the final resultant will be the same regardless of the sequence used. This property is called the commutative law of vector addition.

Tail-to-tail method

In this approach using the tail-to-tail method, all vectors are drawn from the same starting point (origin):

Draw all vectors starting from the origin
From the head of each vector, draw a line parallel to the other vectors
Where these parallel lines intersect is the head of the resultant vector
Draw the resultant from the origin to this intersection point

Working with perpendicular vectors

Perpendicular vectors are vectors that meet at right angles (90°). These are particularly important because they form the basis for component analysis in two dimensions.

Finding components

When dealing with perpendicular vectors, we can separate them into:

Rx: the horizontal component (parallel to x-axis)
Ry: the vertical component (parallel to y-axis)

Using Pythagoras' theorem

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Critical Formula for Perpendicular Vectors

For perpendicular vectors, the magnitude of the resultant can be calculated using Pythagoras' theorem:

$R^2 = R_x^2 + R_y^2$

Where:

$R$ = magnitude of the resultant vector
$R_x$ = magnitude of the horizontal component
$R_y$ = magnitude of the vertical component

Worked example: Finding resultant magnitude

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Worked Example: Calculating Resultant Magnitude

Question: Find the magnitude of the resultant for perpendicular forces of 4 N horizontally and 5.5 N vertically.

Solution:

Step 1: Identify the perpendicular components

$R_x = 4 \text{ N}$ (horizontal)
$R_y = 5.5 \text{ N}$ (vertical)

Step 2: Apply Pythagoras' theorem $R^2 = R_x^2 + R_y^2$ $R^2 = (4)^2 + (5.5)^2$ $R^2 = 16 + 30.25$ $R^2 = 46.25$

Step 3: Calculate the magnitude $R = \sqrt{46.25} = 6.8 \text{ N}$

Answer: The magnitude of the resultant is 6.8 N

Component method for multiple vectors

When dealing with several vectors, you can use this systematic approach:

Find $R_x$ : Add all forces pointing right, subtract all forces pointing left
Find $R_y$ : Add all forces pointing up, subtract all forces pointing down
Calculate resultant magnitude: Use $R = \sqrt{R_x^2 + R_y^2}$

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This method is particularly useful because it reduces complex vector problems to simple arithmetic with perpendicular components. It's often more accurate than purely graphical methods.

Practical applications

Vector analysis is essential in many physics contexts where multiple forces or directional quantities are involved:

Forces acting on objects from different directions
Displacements involving movement in multiple directions
Velocities with both horizontal and vertical components
Accelerations in two-dimensional motion

The principles learned here apply to all vector quantities, not just forces.

Exam tips

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Exam Strategy Tips

Always draw clear diagrams with labelled axes
Use consistent scales when sketching vectors
Remember that perpendicular vectors form right triangles
Check your calculations using both graphical and mathematical methods
Pay attention to direction - positive and negative components matter

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Key Points to Remember:

Vectors have both magnitude and direction, unlike scalar quantities
Perpendicular vectors meet at 90° angles and form the basis for component analysis
Pythagoras' theorem ( $R^2 = R_x^2 + R_y^2$ ) applies only to perpendicular vectors
The head-to-tail method connects vectors end-to-end to find the resultant
Vector equality means same magnitude and direction, regardless of position
Component analysis reduces complex 2D problems to simple arithmetic

Resultant of Perpendicular Vectors (Grade 11 NSC Matric Physical Sciences): Revision Notes