Intersect

Intersect

Have you ever seen two roads crossing each other?

These roads would be intersecting each other.

The figure below shows intersecting roads.

Aerial view of cars at intersection

The topic that we will be learning today in this mini-lesson deals with intersects.

Lesson Plan

What Is an Intersect?

Intersect definition: Intersects can be defined for lines and for sets.

When we talk about an intersect for a line, it would mean to divide by passing through or across something.

Intersection of two lines

However, when we talk about two sets, an intersect would mean if they have at least one element in common.

Intersection of two sets

Here is an interesting simulation to find the intersection of sets:


What is Intersection of Lines?

When two lines or more than two lines cross each other in a plane, this is called the intersection of lines, and the lines are called intersecting lines.

They have a common point called the point of intersection.

In the figure shown below, we can see line P and line Q are intersecting at point O.

point of intersection of two lines.

Cramer's Rule to Find the Point of Intersection

The Cramer's rule is used for solving linear equations.

We can use Cramer’s rule to find out the point of intersection:

\[\begin{align}\frac{x_{0}}{b_{1}c_{2}-b_{2}c_{1}}=\frac{-y_{0}}{a_{1}c_{2}-a_{2}c_{1}}=\frac{1}{a_{1}b_{2}-a_{2}b_{1}}\end{align}\]


What Is Intersection of Sets?

The intersection of two sets says \(A\) and \(B\) is the set that contains all the elements of set \(A\) that also belong to set \(B\). 

It is denoted as:

\(A\) \(\cap\) \(B\)

The below-given image shows two sets, \(A\) and \(B\), and their intersection.

Intersection of Sets


What Is the Intersection Symbol for Sets?

The intersection of sets is represented by the symbol \(\bigcap\).

 
important notes to remember
Important Notes
  1. The intersection of two sets is considered to be the largest set which contains all the common elements of both sets.
  2. Do you know that there are no symbols to represent intersecting lines?
  3. The point of contact of two lines is called the point of intersection.
  4. Intersecting lines can never be parallel.

Solved Examples

Example 1

 

 

If \(A\) = {1, 2, 3} and \(B\) = {2, 3, 4}, can you find \(A\) \(\cap\) \(B\)?

Solution

Given \(A\) = {1, 2, 3} and \(B\) = {2, 3, 4}.

Hence, \(A\) \(\cap\) \(B\) will be the common elements between the two sets: {2, 3}

\(\therefore\) \(A\) \(\cap\) \(B\) = {2, 3}
Example 2

 

 

If \(A\) = {4, 5, 6, 7} , \(B\) = {6, 7, 8, 9}, and \(C\) = {1, 2, 3}, find \(A\) \(\cap\) \(B\) \(\cap\) \(C\).

Solution

Given:

\(A\) = {4, 5, 6, 7}

\(B\) = {6, 7, 8, 9}

\(C\) = {1, 2, 3}

Then, 

A\(\cap\) B \(\cap\) C = {6, 7} \(\cap\) { }

\(\therefore\) A\(\cap\) B \(\cap\) C = {6, 7} \(\cap\) { }
Example 3

 

 

If P and Q are two lines such that P is \(x-2y+3 = 0\) and Q is \(3x-4y+5=0\), and they are intersecting, can you find the point of intersection of P and Q?

Solution

 P is \(x-2y+3 = 0\) and Q is \(3x-4y+5=0\)

Hence, \[\begin{align}a_{1} &= 1, &a_{2} = 3\\
b_{1} &= -2,& b_{2} = -4\\
c_{1} &= 3, &c_{2} = 5\end{align}\]

We will use Cramer’s rule to find the point of intersection:
\[\begin{align}\frac{x_{0}}{b_{1}c_{2}-b_{2}c_{1}}=\frac{-y_{0}}{a_{1}c_{2}-a_{2}c_{1}}=\frac{1}{a_{1}b_{2}-a_{2}b_{1}}\end{align}\]

\[\begin{align}\frac{x}{-10-(-12)}=\frac{-y}{-9+5}=\frac{1}{-4-(-6)}\end{align}\]

\[\begin{align}\frac{x}{2}=\frac{y}{4}=\frac{1}{2}\end{align}\]

We will get x = 1 and y = 2

And the point of intersection (x, y) = (1, 2)

\(\therefore\) (x, y) = (1, 2)
 
 
 
Challenge your math skills
Challenging Questions
  1. If A = {4, 5, 6, 7} , B = {6, 7, 8, 9}, C = {3,5,7}, and D = {1, 2, 3}, then find A\(\cap\) B and C \(\cap\) D.
  2. If two lines P and Q are intersecting such that P = 5x - 4y - 2 and Q = 6x - 5y - 1, then find the point of intersection.

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 

Let's Summarize!

The mini-lesson targeted in the fascinating concept of intersect. The math journey around factors starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it problems, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.


Frequently Asked Questions (FAQs)

1. Do parallel lines intersect?

Parallel lines never intersect as they are on a plane and always have some distance between them because of which no point of contact exists between them.

2. Do two vectors intersect?

Yes, two vectors can intersect.

3. How many points can two lines intersect?

There can be only one point of intersection, but there can be an infinite number of lines passing through it.

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