1. Representing Vectors as Directed Line Segments
A vector is a quantity defined by both magnitude (length) and direction. It is graphically represented by a directed line segment: a straight line with an arrowhead indicating direction.
The length of the line segment corresponds to the vector's magnitude, and the arrow points from the initial point to the terminal point.
For example, a vector AB is drawn as a line from point A to point B with an arrow at B. The vector is denoted as \( \vec{AB} \).
- Vectors have both magnitude and direction.
- Graphically shown as arrows from initial to terminal points.
- Notation: \( \vec{AB} \) represents vector from A to B.
2. Position Vectors and Locating Points
A position vector locates a point relative to a fixed origin O. The position vector of point P is \( \vec{OP} \), the vector from O to P.
Position vectors allow us to describe points in a plane or space using vectors starting at the origin.
For example, if O is at (0,0) and P is at (3,4), then \( \vec{OP} = 3\hat{i} + 4\hat{j} \), where \( \hat{i} \) and \( \hat{j} \) are unit vectors along the x- and y-axes respectively.
- Position vectors start at origin O and end at point P.
- They represent the coordinates of points as vectors.
- Example: \( \vec{OP} = 3\hat{i} + 4\hat{j} \) locates P at (3,4).
3. Expressing Vectors Using Two Coplanar Vectors
Any vector in a plane can be expressed as a sum or difference of two non-parallel base vectors in that plane, often called coplanar vectors.
If \( \vec{a} \) and \( \vec{b} \) are two such vectors, then any vector \( \vec{v} \) in the plane can be written as \( \vec{v} = p\vec{a} + q\vec{b} \), where p and q are scalars.
For example, if \( \vec{a} = \vec{AB} \) and \( \vec{b} = \vec{AC} \) for points A, B, and C not on the same line, then \( \vec{AD} \) for some point D in the plane can be expressed as \( \vec{AD} = p\vec{a} + q\vec{b} \).
- Two non-parallel vectors \( \vec{a} \) and \( \vec{b} \) form a basis for the plane.
- Any vector in the plane can be written as \( p\vec{a} + q\vec{b} \).
- Scalars p and q are found by vector addition and subtraction.
4. Using Vectors to Prove Parallelism and Collinearity
Two vectors are parallel if one is a scalar multiple of the other: \( \vec{u} = k\vec{v} \) for some scalar k.
To prove three points A, B, and C are collinear, show that \( \vec{AB} \) is a scalar multiple of \( \vec{AC} \) or vice versa.
For example, if \( \vec{AB} = 2\vec{AC} \), then points A, B, and C lie on the same straight line.
- Parallel vectors satisfy \( \vec{u} = k\vec{v} \).
- Collinearity of points shown by vector multiples.
- Use vector subtraction to find \( \vec{AB} \) and \( \vec{AC} \).
5. Solving Geometric Problems Involving Ratios and Similarity Using Vectors
Vectors can express points dividing a segment in a given ratio. If point P divides segment AB in ratio m:n, then \( \vec{OP} = \frac{n\vec{OA} + m\vec{OB}}{m+n} \).
This method helps solve problems involving similarity and proportional segments.
For example, if P divides AB in ratio 2:3, then \( \vec{OP} = \frac{3\vec{OA} + 2\vec{OB}}{5} \).
- Use weighted sums of position vectors to find points dividing segments.
- Apply to problems involving ratios and similarity.
- Calculate position vectors using \( \vec{OP} = \frac{n\vec{OA} + m\vec{OB}}{m+n} \).
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Representing vectors | Draw directed line segments with arrow showing direction and length showing magnitude | Vector \( \vec{AB} \) from A(1,2) to B(4,6) | Draw accurately and label vectors clearly in diagrams |
| Position vectors | Express points relative to origin O as \( \vec{OP} \) | Point P(3,4) has \( \vec{OP} = 3\hat{i} + 4\hat{j} \) | Use to find coordinates and vector expressions |
| Expressing vectors | Write any vector as \( p\vec{a} + q\vec{b} \) where \( \vec{a}, \vec{b} \) are base vectors | \( \vec{AD} = 2\vec{AB} - \vec{AC} \) | Find unknown vectors in terms of known vectors |
| Proving parallelism | Show \( \vec{u} = k\vec{v} \) for scalar k | If \( \vec{AB} = 3\vec{CD} \), vectors are parallel | Use scalar multiples to justify parallel lines |
| Proving collinearity | Show vectors between points are scalar multiples | If \( \vec{AB} = 2\vec{AC} \), points A, B, C are collinear | Use vector ratios to prove points lie on a line |
| Ratios and similarity | Find point dividing segment using weighted sum of position vectors | P divides AB in 2:3, \( \vec{OP} = \frac{3\vec{OA} + 2\vec{OB}}{5} \) | Calculate exact position of points dividing segments |
Active recall
Quick check
1. How do you represent a vector graphically?
As a directed line segment with an arrow showing direction and length representing magnitude.
2. What is a position vector?
A vector from the origin to a point, locating that point relative to the origin.
3. How can you express any vector in a plane using two coplanar vectors?
By writing it as a linear combination (sum of scalar multiples) of the two base vectors.
4. How do you prove two vectors are parallel?
Show that one vector is a scalar multiple of the other.
5. What condition shows three points are collinear using vectors?
The vector between two points is a scalar multiple of the vector between another pair of points.