Welcome, dear student! Today we are going to learn a very important topic in physics — Vector Quantities. This is Unit 1 of your Grade 10 Physics textbook. Vectors appear again and again in physics, so understanding them well now will help you a lot in Grade 11 and Grade 12 too. Let’s build a strong foundation together!
In science, we measure things precisely. Any number or set of numbers used to describe a physical phenomenon is called a physical quantity. These physical quantities are divided into two groups: scalars and vectors. In this lesson, we will learn the difference between them, how to represent vectors, how to add and subtract vectors, and how to resolve a vector into components. Everything here is from your Grade 10 Physics textbook, so read carefully and practise the questions!
1.1 Scalars and Vectors
Now, let me ask you a question. Have you ever thought about why we say “5 kg of sugar” but we say “10 Newton of force towards the East”? Why do we need to mention direction for force but not for mass? Well, this is the whole idea behind scalars and vectors. Let me explain this deeply so you never get confused.
What is a Scalar Quantity?
A scalar quantity is a physical quantity that can be completely described by a single number together with an appropriate unit. Nothing else is needed. For example, if I say “the length of this desk is 1.5 metres,” that is a complete description. You do not need to say “1.5 metres to the right” or “1.5 metres upwards.” Length does not need direction. That makes it a scalar.
Think about it this way. When you tell your friend “I walked 3 km,” you are giving a scalar description. You are only telling the distance, not which way you walked. The number plus the unit is enough. That is the key idea of scalars.
Here are common examples of scalar quantities from your textbook:
Time, distance, speed, length, volume, temperature, energy, power, mass, and density
Notice something important. All of these quantities can be fully described by just a number and a unit. For instance, the density of water is 1000 kg/m³. That statement is complete. No direction is needed. The temperature in Addis Ababa is 22°C. Again, complete. No direction. That is what makes them scalars.
What is a Vector Quantity?
Now here is where things get interesting. A vector quantity is a physical quantity that requires both magnitude and direction for a complete description. If you leave out the direction, the description is incomplete.
Let me give you an example from your textbook. Suppose someone tells you “the velocity of a train is 100 km/h.” Does that make full sense? Not really! You would naturally ask: “In which direction?” Is the train going north to Bahir Dar, or south to Hawassa, or east to Dire Dawa? The direction matters a lot. Without it, the description is incomplete. That is why velocity is a vector.
Force is another great example. If I push a box with 20 N of force, you need to know: am I pushing it to the left, to the right, or upwards? The effect of the force depends completely on its direction. So force is a vector quantity.
Displacement, velocity, acceleration, force, momentum, impulse, weight, and electric field strength
Scalars and vectors can NEVER be added together. You cannot add a force to a velocity. Even among vectors, only vectors of the same kind can be added. Two forces can be added. Two velocities can be added. But a force and a velocity cannot be added. This is a very common trick in exams!
| Scalar | Vector |
|---|---|
| Has only magnitude | Has both magnitude and direction |
| Described by number + unit | Described by number + unit + direction |
| Example: mass, speed, time | Example: force, velocity, displacement |
| Added by simple arithmetic | Added by special rules (triangle, parallelogram) |
Practice Questions — Scalars and Vectors
Explanation: Weight is a force. It always acts towards the centre of the Earth (downwards). Since it has both magnitude and direction, weight is a vector quantity. Temperature, volume, and mass are all scalars because they only have magnitude.
Explanation: The student only mentions distance (5 km) and time (30 minutes). No direction is mentioned. When you divide distance by time without considering direction, you get speed, which is a scalar quantity. If the student had said “5 km North in 30 minutes,” then it would be velocity (a vector).
Explanation: This is the fundamental difference. Option A is wrong because scalars and vectors can never be added together. Option B is wrong because only vectors of the same type can be added (e.g., force + force, not force + velocity). Option D is wrong because many quantities with units (like mass in kg) are scalars, not vectors.
1.2 Vector Representation
Now that we know what vectors are, how do we write them down or draw them? This is what we call vector representation. There are two ways to represent vectors: algebraically and geometrically. Both methods are important for your exam.
Algebraic Representation
When we write vectors using letters and symbols, it is called algebraic representation. In your textbook, vectors are represented in two ways:
Method 1: Using a bold letter — for example, A
Method 2: Using an arrow over the letter — for example, A⃗
Both methods mean the same thing: “this is a vector, not just a number.” When you see a bold letter or a letter with an arrow, you should immediately think “magnitude AND direction.”
Now, what about the magnitude only? The magnitude is just the size or numerical value of the vector, without the direction. Your textbook shows us two ways to write the magnitude:
If the vector is A⃗, then its magnitude can be written as:
A or |A⃗|
For example, S⃗ = 50 km, Southwest → The magnitude is S = 50 km (or |S⃗| = 50 km)
See the difference? S⃗ is the full vector (magnitude + direction). But S or |S⃗| is just the magnitude (the number part only). This distinction is very important in exam questions!
Geometric (Graphical) Representation
This is where we draw the vector as an arrow. Your textbook says vectors are represented geometrically by an arrow, or an arrow-tipped line segment. Let me explain each part of this arrow carefully.
So when you draw a vector as an arrow:
- The length of the arrow represents the magnitude (drawn to scale)
- The arrow head shows the direction
- The starting point is called the tail
- The ending point (with the arrow) is called the head
Drawing Vectors to Scale
This is a very practical skill for your exam. You must know how to draw vectors correctly using a ruler and a protractor. Here are the steps from your textbook:
- Decide on a scale and write it down clearly (e.g., 1 cm = 4 km)
- Calculate the arrow length using the scale (e.g., 16 km ÷ 4 km/cm = 4 cm)
- Draw the arrow with the correct length using a ruler
- Fill in the arrow head pointing in the correct direction
- Label the magnitude next to the arrow
Problem: Draw the vector “16 km East” to scale, indicating the scale you used.
Step 1: Choose a scale. Let 1 cm represent 4 km.
Step 2: Calculate length: 16 km ÷ 4 km/cm = 4 cm
Step 3: Draw a horizontal arrow of length 4 cm pointing to the right (East).
Types of Vectors
Your textbook mentions four special types of vectors. You should know all of them for your exam:
| Type of Vector | Definition | Example / Note |
|---|---|---|
| Zero Vector (Null Vector) | A vector with zero magnitude and no specific direction | When you walk from home to school and back home, your total displacement is a zero vector |
| Unit Vector | A vector that has a magnitude of exactly one (1) | Used to show direction only, since the magnitude is just 1 |
| Equal Vectors | Two vectors that have the same magnitude AND the same direction | Both conditions must be met — same size AND same direction |
| Negative of a Vector | A vector that has the same magnitude but opposite direction | If A⃗ is 10 N East, then -A⃗ is 10 N West |
Students often confuse “equal vectors” with “vectors of the same magnitude.” Remember: equal vectors must have BOTH the same magnitude AND the same direction. If two vectors have the same magnitude but different directions, they are NOT equal!
Practice Questions — Vector Representation
Explanation: The scale is 15 mm represents 30 m/s. So 1 mm represents 2 m/s (30 ÷ 15 = 2). For 20 m/s: 20 ÷ 2 = 10 mm. This is a direct question from your textbook’s end-of-unit questions!
Explanation: The negative of a vector has the same magnitude but opposite direction. So -A⃗ has magnitude 8 N but points South (opposite of North). Option D is wrong because “-8 N to the North” does not make physical sense — magnitude is always positive, direction is what changes.
Explanation: By definition from your textbook, equal vectors must have both the same magnitude AND the same direction. Same magnitude alone is not enough. For example, 10 N East and 10 N West have the same magnitude but different directions, so they are NOT equal.
1.3 Vector Addition and Subtraction
Here is a very important question: can you add two vectors the same way you add two numbers? For example, if you have a force of 3 N East and another force of 4 N North, is the total force 7 N? The answer is NO! That would be wrong. Let me explain why, and how to do it correctly.
Why Vectors Are Not Added Like Scalars
When you add scalars, it is simple arithmetic. 2 kg + 3 kg = 5 kg. But vectors have direction, so you cannot just add the numbers. You must consider which way each vector is pointing. Think about it practically: if you walk 3 km East and then 4 km North, are you 7 km away from where you started? No! You are actually only about 5 km away (by Pythagoras theorem). The direction of your final position is somewhere between East and North. This is why we need special rules for adding vectors.
What is a Resultant Vector?
The resultant vector is the single vector that has the same effect as all the individual vectors acting together. In other words, if you replace all the individual vectors with just the resultant vector, the final outcome is exactly the same. We write it as:
R⃗ = A⃗ + B⃗
Where R⃗ is the resultant, A⃗ and B⃗ are the two vectors being added.
Remember: only vectors of the same kind can be added. You can add two forces. You can add two displacements. But you cannot add a force to a displacement.
Vector Subtraction
Vector subtraction is actually very simple once you understand vector addition. To subtract vector B⃗ from vector A⃗, you just flip B⃗ to get -B⃗ (same magnitude, opposite direction), and then ADD -B⃗ to A⃗.
A⃗ − B⃗ = A⃗ + (−B⃗)
Vector subtraction = addition of a negative vector. That is all there is to it! First flip the vector you are subtracting, then add it normally. The order of subtraction DOES matter: A⃗ − B⃗ is NOT the same as B⃗ − A⃗ (they have the same magnitude but opposite directions).
Practice Questions — Vector Addition and Subtraction Basics
Explanation: Force and velocity are different types of vectors. Only vectors of the same kind can be added. You cannot add a force to a velocity, just like you cannot add kilograms to metres. This rule is stated clearly in your textbook.
Explanation: Your textbook defines vector subtraction as: A⃗ − B⃗ = A⃗ + (−B⃗). You flip B⃗ to get −B⃗, and then add it to A⃗. Option C is wrong because vector subtraction is not simply subtracting the magnitudes — direction matters!
1.4 Graphical Method of Vector Addition
Now we come to one of the most important sections for your exam. How do we actually add vectors in practice? In Grade 10, you learn the graphical methods of vector addition. There are three methods: the Triangle Method, the Parallelogram Method, and the Polygon Method. Let me teach each one deeply.
General Procedure for All Graphical Methods
Before we look at each specific method, here is the common procedure you must follow for any graphical method. Your textbook gives these steps clearly:
- Decide on a scale — choose one that gives a diagram of reasonable size
- Record the scale on your diagram
- Pick a starting point
- Draw the first vector with the correct length and direction
- Draw the second and remaining vectors with correct length and direction
- Draw the resultant based on the specific rule you are using
- Measure the resultant’s length with a ruler and convert using the scale
- Measure the resultant’s direction with a protractor
1. Triangle Method of Vector Addition
The triangle method is also called the head-to-tail method. This is probably the most commonly used method in exams. Here is the idea: you place the tail of the second vector at the head of the first vector. Then the resultant is drawn from the tail of the first vector to the head of the second vector. The three vectors form a triangle.
Let me explain this step by step. You have two vectors A⃗ and B⃗. You first draw A⃗ starting from some point O. The head of A⃗ becomes the tail of B⃗, so you draw B⃗ starting where A⃗ ends. Now, the resultant R⃗ is the vector that goes from the tail of A⃗ (point O) to the head of B⃗. It is like completing the triangle.
Your textbook states the triangle law clearly: “If two vectors acting simultaneously on a body are represented both in magnitude and direction by two sides of a triangle taken in an order, then the resultant vector is given by the third side of that triangle taken in the opposite order.”
2. Parallelogram Method of Vector Addition
The parallelogram method is another way to add two vectors. In this method, both vectors start from the same point (their tails are joined together). Then you complete a parallelogram, and the diagonal gives the resultant.
Here is how it works. You have vectors A⃗ and B⃗. You place both of them so their tails are at the same point O. Then, from the head of A⃗, you draw a line parallel to B⃗. From the head of B⃗, you draw a line parallel to A⃗. These two lines meet at a point, forming a parallelogram. The diagonal from O to the meeting point is your resultant R⃗.
Your textbook states: “If two vectors are represented by the two adjacent sides of a parallelogram drawn from a point, then their resultant vector is represented by the diagonal of the parallelogram drawn from the same point.”
3. Polygon Method of Vector Addition
What if you have more than two vectors? This is where the polygon method becomes useful. The polygon method is basically an extension of the triangle method. You keep placing vectors head-to-tail, one after another. The resultant is drawn from the tail of the first vector to the head of the last vector.
The procedure is simple. Take any vector as the first one. Place the tail of the second vector at the head of the first. Then place the tail of the third vector at the head of the second. Keep going until all vectors are placed head-to-tail. Finally, draw the resultant from the tail of the first vector to the head of the last vector.
Because vector addition is both commutative and associative, you get the same resultant regardless of the order in which you arrange the vectors. You can start with any vector and still arrive at the same answer.
Special Cases of Vector Addition
Now pay very close attention! These three special cases appear very frequently in exams. Your textbook discusses them in detail, and you must know all three.
Special Case 1: Two Vectors in the Same Direction (Parallel)
When two vectors point in the same direction, the resultant is very easy to find. You simply add their magnitudes, and the direction of the resultant is the same as the direction of both vectors.
|R⃗| = |A⃗| + |B⃗|
Direction = same direction as A⃗ and B⃗
Special Case 2: Two Vectors in Opposite Directions (Anti-parallel)
When two vectors point in opposite directions, you subtract the smaller magnitude from the larger magnitude. The direction of the resultant is the direction of the vector with the larger magnitude.
|R⃗| = |A⃗| − |B⃗| (assuming A is larger)
Direction = direction of the larger vector
Special Case 3: Two Perpendicular Vectors
When two vectors are at right angles (90°) to each other, we use the Pythagorean theorem to find the magnitude of the resultant, and trigonometry (tangent) to find its direction.
|R⃗| = √(A² + B²)
θ = tan⊃¹(B / A)
The maximum resultant of two vectors occurs when they are in the same direction: Rmax = A + B
The minimum resultant of two vectors occurs when they are in opposite directions: Rmin = |A − B|
If two vectors have equal magnitudes and are in opposite directions, the resultant is zero (a null vector).
Problem: Two vectors have magnitudes of 6 units and 3 units. What is the magnitude of the resultant vector when the two vectors are (a) in the same direction, (b) in opposite direction, and (c) perpendicular to each other?
(a) Same direction:
|R⃗| = 6 + 3 = 9 units
(b) Opposite direction:
|R⃗| = 6 − 3 = 3 units
(Direction is that of the 6-unit vector, since it is larger)
(c) Perpendicular:
|R⃗| = √(6² + 3²) = √(36 + 9) = √45 = 6.7 units
Practice Questions — Graphical Method of Vector Addition
Explanation: Since B⃗ is exactly opposite to A⃗ (South-West is opposite to North-East) and both have the same magnitude (5 units), they cancel each other out completely. This is the minimum resultant case: Rmin = |5 − 5| = 0. The resultant is a null vector (zero vector). This question is directly from your textbook’s review questions!
Explanation: The two forces are perpendicular (one in x-direction, one in y-direction). So we use the Pythagorean theorem: |R⃗| = √(6² + 8²) = √(36 + 64) = √100 = 10 N. This is a classic 3-4-5 triangle scaled up (6-8-10). This question is based on your textbook’s review question about vector C⃗ (6 m x-direction) and D⃗ (8 m y-direction).
Explanation: If the head of the last vector meets the tail of the first vector, it means the vectors form a closed polygon. In the polygon method, the resultant is drawn from the tail of the first to the head of the last. If they are at the same point, the resultant has zero length. This is a null vector. This question is directly from your textbook’s review questions!
Explanation: Three vectors can sum to zero if they form a closed triangle when placed head-to-tail. This is possible even with unequal magnitudes. The only requirement is that no single magnitude is greater than the sum of the other two (triangle inequality rule). For example, vectors of 3, 4, and 5 units can form a closed triangle (3² + 4² = 5²), so their sum is zero. This is from your textbook’s review question.
1.5 Vector Resolution
We have learned how to combine vectors to get a resultant. Now we are going to do the reverse. We are going to take one vector and break it into smaller vectors that, when added together, give back the original vector. This process is called vector resolution. It is a very powerful technique and is heavily tested in exams.
What is Vector Resolution?
Imagine you throw a ball at an angle. The ball moves both forward and upward at the same time. The single velocity of the ball can be thought of as having two parts: a horizontal part (moving forward) and a vertical part (moving up). These two parts are called the components of the velocity vector.
So, vector resolution is the process of breaking a single vector into two (or more) component vectors that add up to the original vector. The most common type of resolution is breaking a vector into horizontal (x) and vertical (y) components.
Trigonometric Method of Vector Resolution
This is the method you will use most often in exams. It uses basic trigonometry: sine and cosine functions. Here is how it works.
Suppose vector A⃗ makes an angle θ with the positive x-axis. Then:
Ax = A cos θ
Ay = A sin θ
Where:
Ax = horizontal (x) component
Ay = vertical (y) component
A = magnitude of the original vector
θ = angle measured from the positive x-axis
COSINE goes with the ADJACENT side (x-component, which is next to the angle θ)
SINE goes with the OPPOSITE side (y-component, which is across from the angle θ)
Think: “Cos = Close (adjacent/x)” and “Sin = Sky (opposite/y, going up)”
And if you already know the components and want to find the original vector, you use these reverse formulas:
|A⃗| = √(Ax² + Ay²)
θ = tan⊃¹(Ay / Ax)
The first formula is just the Pythagorean theorem — because the x and y components are perpendicular to each other, they form a right triangle with the original vector as the hypotenuse. The second formula uses the tangent function to find the angle.
Problem: A motorist undergoes a displacement of 250 km in a direction 30° North of East. Resolve this displacement into its components.
The displacement makes an angle of 30° with the East direction (which is our x-axis).
North component (y-component):
SN = 250 × sin 30° = 250 × 0.5 = 125 km
East component (x-component):
SE = 250 × cos 30° = 250 × 0.866 = 216.5 km
So the motorist moved 216.5 km East and 125 km North. Together, these give the original displacement of 250 km at 30° North of East.
Problem: A boy walks 3 km due East and then 2 km due North. What is the magnitude and direction of his displacement vector?
Here, the East walk (3 km) is the x-component and the North walk (2 km) is the y-component of the displacement.
Magnitude of displacement:
|S⃗| = √(3² + 2²) = √(9 + 4) = √13 = 3.61 km
Direction (angle from East):
tan θ = 2 / 3 = 0.666
θ = tan⊃¹(0.666) = 33.69°
So the displacement is 3.61 km at 33.69° North of East (which can also be expressed as 56.31° East of North).
Graphical Method of Vector Resolution
Your textbook also explains how to resolve a vector graphically using a ruler and protractor. Here are the steps:
- Select a scale and draw the vector to scale in the correct direction
- Extend x- and y-axes from the tail of the vector
- From the arrow head of the vector, draw perpendicular lines (dashed lines) to the x-axis and y-axis
- The x-component is the distance from the tail to where the perpendicular meets the x-axis
- The y-component is the distance from the tail to where the perpendicular meets the y-axis
- Measure both components with a ruler and convert using your scale
Always check which angle is given! Sometimes the angle is measured from the x-axis (East), and sometimes from the y-axis (North). For example, “30° North of East” means θ = 30° from East (x-axis), so you use cos 30° for the East component and sin 30° for the North component. But “30° East of North” means the angle is from North (y-axis), so you would use cos 30° for North and sin 30° for East. Read the wording carefully!
Practice Questions — Vector Resolution
Explanation: The horizontal component is Fx = F cos θ = 40 × cos 30° = 40 × 0.866 = 34.6 N. We use cosine because the horizontal component is the adjacent side to the 30° angle. This is based on your textbook’s review question about resolving 40 N at 30° from the horizontal.
Explanation: The North component (how far due North) = 3.10 × sin 25° = 3.10 × 0.4226 = 1.31 km. We use sine because the North component is opposite to the 25° angle (measured from East). The East component would be 3.10 × cos 25° = 2.81 km. This question is directly from your textbook’s review questions!
Explanation: Using the Pythagorean theorem: |V⃗| = √(Vx² + Vy²) = √(9.8² + 6.4²) = √(96.04 + 40.96) = √137 = 11.7 m/s. This is a direct question from your textbook’s end-of-unit questions! Also, the direction would be θ = tan⊃¹(6.4/9.8) = 33.1° from the x-axis.
Explanation: The North component = 10 km and the West component = 5 km. Since these are perpendicular, |R⃗| = √(10² + 5²) = √(100 + 25) = √125 = 11.2 km. The direction is θ = tan⊃¹(5/10) = 26.6° West of North. This is from your textbook’s review question.
End of Unit Summary
Congratulations! You have completed Unit 1 on Vector Quantities. Let me now give you a quick summary of everything you need to remember for your exam. Read this carefully before you walk into the exam hall.
1. Same direction: |R⃗| = |A⃗| + |B⃗|
2. Opposite direction: |R⃗| = ||A⃗| − |B⃗||
3. Perpendicular vectors: |R⃗| = √(A² + B²)
4. Direction of perpendicular resultant: θ = tan⊃¹(B/A)
5. Horizontal component: Ax = A cos θ
6. Vertical component: Ay = A sin θ
7. Magnitude from components: |A⃗| = √(Ax² + Ay²)
8. Direction from components: θ = tan⊃¹(Ay / Ax)
9. Vector subtraction: A⃗ − B⃗ = A⃗ + (−B⃗)
- Scalar = magnitude only (e.g., mass, speed, time, temperature)
- Vector = magnitude + direction (e.g., force, velocity, displacement, weight)
- Vectors are drawn as arrows where length = magnitude and arrow head = direction
- Always draw vectors to scale and label the scale on your diagram
- Triangle method: place vectors head-to-tail, resultant from first tail to last head
- Parallelogram method: both tails at same point, resultant is the diagonal
- Polygon method: for 3+ vectors, keep placing head-to-tail
- Resultant is maximum when vectors are in the same direction
- Resultant is minimum when vectors are in opposite directions
- Vector resolution = breaking a vector into x and y components using sin and cos
- Scalars and vectors can NEVER be added together
- Only vectors of the same type can be added (force + force, not force + velocity)
✓ Always write the scale when drawing vector diagrams — marks are allocated for this!
✓ Always use a ruler for measuring lengths and a protractor for measuring angles in graphical methods
✓ For vector resolution, carefully identify which angle is given and from which reference direction
✓ Remember: cos = adjacent (x-component), sin = opposite (y-component)
✓ In MCQs, check whether the question asks for a scalar or a vector answer
✓ The resultant of a closed polygon is always zero
✓ When two equal vectors are opposite, the resultant is a null vector