8.1 Vectors
In our daily lives, quantities can either involve magnitude only or both magnitude and direction. Understanding the difference between scalar and vector quantities is essential in Additional Mathematics and physics. This topic introduces vector notations, directed line segments, zero vectors, negative vectors, equal vectors, scalar multiplication, and the conditions for two vectors to be parallel or for three points to be collinear.
Learning Objectives
- Differentiate between scalar quantities and vector quantities with real-life examples.
- Represent vectors using directed line segments and standard notations, and determine their magnitude and direction.
- Identify zero vectors, equal vectors, and negative vectors.
- Perform and explain scalar multiplication on vectors, including effects on magnitude and direction.
- Apply the condition for two parallel vectors and solve problems involving non-parallel non-zero vectors.
- Prove and apply the conditions for collinear points using vector relations with a common point.
Scalar vs Vector Quantities & Vector Properties Summary
Compare scalar and vector pairs, then explore vector notation, negative vectors, scalar multiplication and collinearity.
Scalar Quantities
Magnitude only, no direction. Examples: distance, speed, mass, temperature, area, volume, density and duration.
Vector Quantities
Magnitude and direction. Examples: displacement, velocity, weight, force, acceleration and impulse.
| Scalar | Vector | Key difference |
|---|---|---|
| Distance | Displacement | Distance is total path length; displacement is the shortest directed line from start to end. |
| Speed | Velocity | Speed is rate of change of distance; velocity is rate of change of displacement with direction. |
| Mass | Weight | Mass is constant everywhere; weight is the gravitational pull and depends on location. |
Negative vector
\(\vec{BA} = -\vec{AB}\)Same magnitude, opposite direction.
Scalar multiplication
\(|k\underset{\sim}{a}| = |k||\underset{\sim}{a}|\)Direction reverses when \(k < 0\).
Short Explanation
Scalar and vector quantities, vector notation, scalar multiplication, parallel vectors and collinear points.
1. Scalar Quantity
A quantity that has magnitude but no direction. Examples: temperature \(-12^\circ\text{C}\), height \(1.48\text{ m}\), area \(120\text{ cm}^2\), density \(1.2\text{ g cm}^{-3}\), volume \(1.5\text{ }l\), duration \(3\text{ days and }2\text{ nights}\).
2. Vector Quantity
A quantity that has both magnitude and direction. Examples: force \(50\text{ N}\) applied to a box, velocity \(90\text{ km h}^{-1}\) heading east, car moving south at \(80\text{ km h}^{-1}\), horizontal impulse of \(5.0\text{ Ns}\).
3. Vector Representation
A vector is represented by a directed line segment with an arrow. The arrow shows direction and the length represents magnitude according to a scale. Notations: \(\vec{AB}\), \(\mathbf{AB}\), \(\underset{\sim}{a}\), or \(\mathbf{a}\).
4. Magnitude of a Vector
Written as \(|\vec{AB}|\), \(|\mathbf{AB}|\), \(|\underset{\sim}{a}|\), or \(|a|\). On a Cartesian grid, \(|\vec{AB}| = \sqrt{x^2 + y^2}\).
5. Zero, Equal and Negative Vectors
Zero vector \(\underset{\sim}{0}\) has zero magnitude and undetermined direction. Equal vectors have the same magnitude and direction. The negative vector of \(\vec{AB}\) is \(\vec{BA} = -\vec{AB}\).
6. Scalar Multiplication
For scalar \(k\), vector \(k\underset{\sim}{a}\) has magnitude \(|k||\underset{\sim}{a}|\). If \(k > 0\), direction is the same; if \(k < 0\), direction is opposite.
7. Parallel Vectors
Two non-zero vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) are parallel if and only if \(\underset{\sim}{a} = k\underset{\sim}{b}\), where \(k\) is a non-zero constant.
8. Collinear Points
Points \(A\), \(B\) and \(C\) are collinear if \(\vec{AB} = k\vec{BC}\) and point \(B\) is a common point.
Non-parallel non-zero vectors
\(h\underset{\sim}{a} = k\underset{\sim}{b} \implies h = 0 \text{ and } k = 0\)This property is used to compare coefficients of non-parallel vectors.
Try to Answer First
Answer in your mind, then press "Check Answer".
Given that \(\vec{AB} = 5\underset{\sim}{a}\) and \(\vec{PQ} = 20\underset{\sim}{a}\), express \(\vec{AB}\) in terms of \(\vec{PQ}\).
Given non-zero and non-parallel vectors \(\underset{\sim}{u}\) and \(\underset{\sim}{v}\), find the values of \(m\) and \(n\) if \((4m + 3)\underset{\sim}{u} = (n - 7)\underset{\sim}{v}\).
Show that points \(L\), \(M\), and \(N\) are collinear if \(\vec{LM} = 6\underset{\sim}{x}\) and \(\vec{MN} = 18\underset{\sim}{x}\).
Common Mistakes
- Confusing scalar and vector counterparts: treating distance as displacement, speed as velocity, or mass as weight.
- Forgetting to write the vector tilde \(\underset{\sim}{a}\) or arrow \(\vec{AB}\) when writing vector variables by hand.
- Omitting the negative sign when writing a vector in the reverse direction, for example writing \(\vec{BA} = \vec{AB}\) instead of \(\vec{BA} = -\vec{AB}\).
- Forgetting the collinearity condition requires both parallelism \(\vec{AB} = k\vec{BC}\) AND stating the existence of a common point.
Concept Misunderstandings
A negative vector has a negative magnitude.
Magnitude is always a non-negative scalar value representing length, \(|\vec{AB}| \ge 0\). The negative sign in \(-\vec{AB}\) indicates the opposite direction, not a negative length.
Mass changes depending on the gravitational field.
Mass is a scalar quantity measuring the amount of matter and remains constant everywhere, for example \(120\text{ kg}\) on Earth and Moon. Weight is the gravitational force, which varies depending on gravity.
Two vectors are parallel only if they have the same magnitude.
Parallel vectors need only have the same or exactly opposite direction, \(\vec{a} = k\vec{b}\). Their magnitudes do not need to be equal.
Summary
- Scalar: Magnitude only — distance, speed, mass, time, temperature.
- Vector: Magnitude and direction — displacement, velocity, weight, force, acceleration.
- Notation: Vector \(\vec{AB}\) or \(\underset{\sim}{a}\); magnitude \(|\vec{AB}|\) or \(|\underset{\sim}{a}|\).
- Zero vector \(\underset{\sim}{0}\): Magnitude = 0, direction undetermined.
- Negative vector: \(\vec{BA} = -\vec{AB}\).
- Scalar multiplication \(k\underset{\sim}{a}\): Same direction if \(k > 0\), opposite direction if \(k < 0\), magnitude \(|k||\underset{\sim}{a}|\).
- Parallel vectors: \(\underset{\sim}{a} = k\underset{\sim}{b}\), where \(k \ne 0\).
- Collinear points: \(\vec{AB} = k\vec{BC}\) with common point \(B\).
- Non-parallel non-zero vectors: \(h\underset{\sim}{a} = k\underset{\sim}{b} \implies h = 0\) and \(k = 0\).
Short Activity
Objective: Identify vector quantities, apply non-parallel vector conditions, and match vector concepts to their descriptions.
A. Objective Quiz
1 Which of the following is a vector quantity?
2 Vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) are non-zero and not parallel. Given that \((h - 1)\underset{\sim}{a} = (k + 5)\underset{\sim}{b}\), find the values of \(h\) and \(k\).
B. Fill in the Blanks
3 If two non-zero vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) are parallel, then \(\underset{\sim}{a} = k\underset{\sim}{b}\), where \(k\) is a ________.
4 The negative vector of \(\vec{AB}\) has the same magnitude but acts in the opposite direction and is written as ________.
C. Matching / Drag and Drop
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
