Vector Proof (H) QP focuses on geometric principles using vector methods. It includes problems related to points A, B, and C lying on a straight line, as well as the relationships between points D, E, and F. This resource is ideal for students studying geometry and vectors in high school mathematics. It provides clear examples and proofs to help understand the concepts of vector alignment and parallelism. The document is suitable for exam preparation and enhances comprehension of geometric proofs.

Key Points

  • Proves that points A, B, and C are collinear using vector equations.
  • Calculates the ratio of lengths DF to DE based on vector relationships.
  • Demonstrates how to use vector methods to show parallel lines in geometry.
  • Includes practical examples relevant for high school geometry students.
Yukta Peerannawar
2 pages
Language:English
Type:Worksheet
Biology
Yukta Peerannawar
2 pages
Language:English
Type:Worksheet
Biology
59
/ 2
*P66305A01624*
16
1 A, B and C are three points such that
AB
→
= 3a + 4b
AC
→
= 15a + 20b
(a) Prove that A, B and C lie on a straight line.
(2)
D, E and F are three points on a straight line such that
DE
→
= 3e + 6f
EF
→
= −10.5e − 21f
(b) Find the ratio
length of DF : length of DE
.......................................................
(3)
(Total for Question 1
is 5 marks)
PMT
Vector Proof (H) - Geometry and Measures
PhysicsAndMathsTutor.com
*P68725A02024*
20
2 CDEF is a quadrilateral.
C
D
E
F
b
2a
a
FE
→
= a ED
→
= b CD
→
= 2a
The point P is such that CEP is a straight line and that CE = EP
Use a vector method to prove that CF is parallel to DP.
(Total for Question 2 is 4 marks)
PMT
Vector Proof (H) - Geometry and Measures
PhysicsAndMathsTutor.com
/ 2
End of Document
59

You May Also Like

FAQs

How can I prove that points A, B, and C are collinear using vectors?
To prove that points A, B, and C lie on a straight line, we can use the vectors AB and AC. Given that AB = 3a + 4b and AC = 15a + 20b, we can express AC as a scalar multiple of AB. Specifically, we can factor out 5 from AC, showing that AC = 5(3a + 4b) = 5AB. Since one vector is a scalar multiple of the other, A, B, and C are collinear.

What is the ratio of the lengths of DF to DE in the line segment problem?
In the problem involving points D, E, and F, we have DE = 3e + 6f and EF = -10.5e - 21f. To find the ratio of the lengths DF to DE, we first express DF in terms of DE and EF. The length of DE can be calculated as the magnitude of the vector DE, which is √((3)^2 + (6)^2) = √(45) = 3√5. Similarly, the length of DF can be determined using the vector representation. After simplification, the ratio length of DF to length of DE is found to be 7:3.

How can vector methods be used to show CF is parallel to DP?
To prove that CF is parallel to DP in quadrilateral CDEF, we start with the vectors FE = a, ED = b, and CD = 2a. The point P is defined such that CEP is a straight line and CE = EP. By expressing the vector CF in terms of the other vectors and using the relationship between CE and EP, we can demonstrate that CF is a scalar multiple of DP. This confirms that CF is parallel to DP, as parallel vectors maintain a constant direction.