Vector Geometry
7.4 Vector Geometry – Cheat Sheet
1. Representing Vectors
- Vectors are represented by directed line segments with both magnitude and direction.
- If vector AB is represented as:
[ x ]
[ y ]
then it goes x units horizontally and y units vertically from point A to point B.
2. Position Vectors
- Position vector of point P(x, y) is:
OP =
[ x ]
[ y ]
- Origin O(0, 0) is always the starting point for a position vector.
3. Vector Addition and Subtraction
Rule:
If
a =
[ x₁ ]
[ y₁ ]
and
b =
[ x₂ ]
[ y₂ ]
then:
| Operation | Formula |
|---|---|
| Addition | a + b = [ x₁ + x₂ ] [ y₁ + y₂ ] |
| Subtraction | a – b = [ x₁ – x₂ ] [ y₁ – y₂ ] |
| Scalar multiplication | k·a = [ kx₁ ] [ ky₁ ] |
4. Expressing Vectors in Terms of Two Coplanar Vectors
- If OA = a and OB = b, any point P in the plane can be expressed as:
OP = m·a + n·b
where m and n are scalars.
5. Showing Vectors are Parallel
- Two vectors are parallel if one is a scalar multiple of the other:
a = k·b
Example:
[ 6 ]
[ 9 ]
is parallel to
[ 2 ]
[ 3 ]
since k = 3.
6. Showing Three Points are Collinear
- For points A, B, C, if vectors AB and AC are parallel, then A, B, C are collinear.
Steps:
- Find AB = B – A
- Find AC = C – A
- Check if AB = k·AC
7. Ratios in Vector Problems
- If point P divides line segment AB in ratio m:n:
OP = (n·OA + m·OB) / (m + n)
| Example | OA = [ 2 ] [ 3 ], OB = [ 8 ] [ 7 ], ratio 1:2 |
|---|---|
| Formula | OP = (2·[ 2 ] [ 3 ] + 1·[ 8 ] [ 7 ]) / (1+2) |
| Working | OP = ( [ 4 ] [ 6 ] + [ 8 ] [ 7 ]) / 3 |
| OP = [ 12 ] [ 13 ] / 3 | |
| Result | OP = [ 4 ] [ 13/3 ] |
8. Solving Similarity Problems with Vectors
- Use parallelism and ratios to find unknown coordinates or vector lengths.
- Example: If triangles OAB and OCD are similar, then OA/OC = OB/OD.