Vectors in two dimensions
Section: Transformations and vectors | Syllabus: Cambridge IGCSE Mathematics (0580)
Vector Notation
- A vector has both magnitude (size) and direction, unlike a scalar, which only has size
- A vector can be written as a column vector x. y , where the top number is the horizontal component and the bottom number is the vertical component
- A vector from point A to point B is written AB, and can also be labelled with a single bold (or underlined) letter, e.g. a
- To find AB from coordinates, subtract the coordinates of the start point from the coordinates of the end point: finish minus start
- Two vectors are equal if they have the same magnitude and the same direction, regardless of where they are drawn
Worked Example: Finding a Vector from Coordinates
- Question: Point C has coordinates (2,5) and point D has coordinates (9,1). Find CD as a column vector.
- Step 1: Subtract the coordinates of C (start) from D (finish): horizontal component =9-2=7
- Step 2: Vertical component =1-5=-4
- Answer: CD= 7. -4
Common Mistakes
MistakeSubtracting the finish point's coordinates from the start point's, giving the reverse vector by mistake
Fixfor the vector "AB", always work out B minus A (finish minus start) - reversing the order gives BA instead, which points the opposite way
MistakeTreating a column vector like a coordinate pair and writing it with brackets and a comma, e.g. (7, -4)
Fixa column vector is written stacked vertically inside large brackets, not side by side like a coordinate
Adding, Subtracting and Scaling Vectors
- To add or subtract column vectors, add or subtract the corresponding top and bottom components separately
- To multiply a vector by a scalar k, multiply both components by k; this scales the vector's length by a factor of k without changing its direction (or reversing it, if k is negative)
- Two vectors are parallel if one is a scalar multiple of the other, e.g. 6. 9 is parallel to 2. 3 since it equals 3 2. 3
- Vectors are often combined using letters, e.g. if a= 4. 1 and b= -2. 3 , then 2a+b= 8. 2 + -2. 3 = 6. 5
Worked Example: Combining Vectors
- Question: p= 5. -2 and q= -1. 4 . Find 3p-2q as a column vector.
- Step 1: Scale p: 3p= 15. -6
- Step 2: Scale q: 2q= -2. 8
- Step 3: Subtract component-wise: 15. -6 - -2. 8 = 17. -14
- Answer: 17. -14
Common Mistakes
MistakeMultiplying only the top component by the scalar and forgetting the bottom component
Fixa scalar multiplies BOTH components of a column vector, top and bottom
MistakeAssuming two vectors are parallel just because they "look similar", without checking one is an exact scalar multiple of the other
Fixto prove vectors are parallel, show algebraically that one equals a number times the other in both components
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