Vectors - Problems - Classifying Quadrilateral Based On Position Vectors

Vectors - Problems - Classifying Quadrilaterals based on Position Vectors

In this lecture, we will discuss how to classify quadrilaterals based on their position vectors in a coordinate plane.
Quadrilaterals are polygons with four sides.
The position vectors of the vertices of a quadrilateral can provide information about its shape and properties.
Let’s consider some examples to understand this concept better.

Example 1

Consider a quadrilateral ABCD with position vectors A(2, 4), B(5, 1), C(8, -2), and D(3, -5). Properties:

A quadrilateral is a parallelogram if and only if the position vectors of opposite sides are equal.
If AB = CD and BC = AD, then the quadrilateral is a parallelogram. Determining Parallelogram:
Calculating the position vectors of the sides:
- AB = (5-2, 1-4) = (3, -3)
- BC = (8-5, -2-1) = (3, -3)
- CD = (3-8, -5+2) = (-5, -3)
- AD = (3-2, -5-4) = (1, -9) Continued on the next slide…

Example 1 (Continued)

Comparing the position vectors of opposite sides:
- AB = CD, BC = AD, so the quadrilateral ABCD is a parallelogram. Properties of Parallelogram:
Opposite sides are parallel.
Opposite angles are equal.
Diagonals bisect each other.
The sum of adjacent angles is 180°.

Example 2

Consider a quadrilateral PQRS with position vectors P(-3, -2), Q(1, -4), R(4, -1), and S(0, 1). Properties:

A quadrilateral is a rectangle if it is a parallelogram and has right angles.
If the dot product of the position vectors of adjacent sides is zero, then the quadrilateral is a rectangle. Determining Rectangle:
Calculating the position vectors of the sides:
- PQ = (1-(-3), -4-(-2)) = (4, -2)
- QR = (4-1, -1-(-4)) = (3, 3)
- RS = (0-4, 1-(-1)) = (-4, 2)
- SP = (-3-0, -2-1) = (-3, -3) Continued on the next slide…

Example 2 (Continued)

Calculating the dot product of adjacent sides:
- PQ · QR = (4 * 3) + (-2 * 3) = 12 - 6 = 6, which is not zero.
  - PQ · QR ≠ 0, so the quadrilateral PQRS is not a rectangle. Conclusion:
The given quadrilateral is a parallelogram but not a rectangle as the dot product of adjacent sides is not zero.

Example 3

Consider a quadrilateral MNOP with position vectors M(1, 3), N(-2, 6), O(-5, 3), and P(-2, 0). Properties:

A quadrilateral is a square if it is a parallelogram, has right angles, and all sides are equal in length.
If the position vectors of opposite sides are equal and the dot product of adjacent sides is zero, then the quadrilateral is a square. Determining Square:
Calculating the position vectors of the sides:
- MN = (-2-1, 6-3) = (-3, 3)
- NO = (-5-(-2), 3-6) = (-3, -3)
- OP = (-2-(-5), 0-3) = (3, -3)
- PM = (1-(-2), 3-0) = (3, 3) Continued on the next slide…

Example 3 (Continued)

Comparing the position vectors of opposite sides:
- MN = OP, NO = PM, so the quadrilateral MNOP is a parallelogram.
Calculating the dot product of adjacent sides:
- MN · NO = (-3 * -3) + (3 * -3) = 9 - 9 = 0
  - MN · NO = 0, so the quadrilateral MNOP is a rectangle. Conclusion:
The given quadrilateral is a rectangle and also a parallelogram. Therefore, it is a square as well, as all four sides are equal.

Summary

The position vectors of a quadrilateral’s vertices provide valuable information about its shape and properties.
By comparing the position vectors of opposite sides, we can determine if the quadrilateral is a parallelogram.
Additionally, the dot product of adjacent sides helps classify the quadrilateral as a rectangle or square.
Understanding these concepts can assist in solving problems related to classifying quadrilaterals based on position vectors.

This concludes the lecture on “Classifying Quadrilaterals based on Position Vectors”. Next, we will dive into more examples and concepts related to vectors.

Slide 11

Let’s continue with more examples and concepts related to classifying quadrilaterals based on position vectors.
Understanding these concepts can help us solve various problems in coordinate geometry.
We will now explore the properties of a kite and a trapezium based on their position vectors.

Slide 12

Properties of a Kite:

A kite is a quadrilateral with two pairs of adjacent sides that are equal in length.
The diagonals of a kite are perpendicular and bisect each other.

Slide 13

Example 4: Consider a quadrilateral KITE with position vectors K(-1, 2), I(3, 1), T(1, -3), and E(-3, -2). Determining Kite:

Calculating the position vectors of the sides:
- KI = (3-(-1), 1-2) = (4, -1)
- IT = (1-3, -3-1) = (-2, -4)
- TE = (-3-1, -2-(-3)) = (-4, 1)
- EK = (-1-(-3), 2-(-2)) = (2, 4)

Slide 14

Example 4 (Continued):

Comparing the position vectors of opposite sides:
- KI = TE, IT = EK, so the quadrilateral KITE is a kite.

Slide 15

Properties of a Trapezium (Trapezoid):

A trapezium is a quadrilateral with one pair of opposite sides that are parallel.
The other two sides are not parallel.

Slide 16

Example 5: Consider a quadrilateral TRAP with position vectors T(-2, -1), R(2, 4), A(5, 4), and P(0, 1). Determining Trapezium:

Calculating the position vectors of the sides:
- TR = (2-(-2), 4-(-1)) = (4, 5)
- RA = (5-2, 4-4) = (3, 0)
- AP = (0-5, 1-4) = (-5, -3)
- PT = (-2-0, -1-1) = (-2, -2)

Slide 17

Example 5 (Continued):

Comparing the position vectors of opposite sides:
- TR ≠ AP, so the quadrilateral TRAP is not a trapezium.

Slide 18

Summary:

The position vectors of a quadrilateral’s vertices help us analyze its properties and shape.
By comparing the position vectors of opposite sides, we can classify quadrilaterals as kites or trapeziums.
Understanding these properties allows us to solve problems involving these types of quadrilaterals in coordinate geometry.

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