Mathematics · Simulation and virtual lab Free
Vectors: addition, subtraction, scalar multiples and the dot product
Tap the grid to set two vectors, then add, subtract, scale them and find the angle between them.
- You will learn to
- Add, subtract and scale vectors using components and arrows.
- Calculate magnitudes and the dot product.
- Find the angle between vectors and test for parallel and perpendicular vectors.
- You will measure
- Components and magnitudes of a, b and the result; a · b and θ.
- Time needed
- About 25–35 min
- Equipment
- Nothing extra. Real version: squared paper, ruler, protractor.
- Price
- Free: no payment, no account.
Virtual lab
Follow the steps with the simulation above (go back to the simulation ↑). Each time you tap Record, a row is added to the table and a point to the graph.
Aim
To represent vectors as column vectors and arrows, to add and subtract vectors and multiply them by a scalar, to calculate magnitudes, and to use the dot product to find the angle between two vectors and test for perpendicular and parallel vectors.
What to do
- Choose a + b. Tap the grid to set a = (4, 2) and b = (1, 5). Check a + b by adding the components. Record.
- Calculate |a|, |b| and |a + b|. Is |a + b| equal to |a| + |b|? When could it be?
- Choose a − b. Where does the arrow a − b start and end?
- Choose k × a. Try k = 2, −1 and 0.5. What do you notice about the direction?
- Choose a · b and the angle. Tap Make b perpendicular to a. What is a · b?
- Use New vectors several times. Predict whether θ is acute or obtuse from the sign of a · b, then check.
Results
Questions
Find a + b when a = (3, −1) and b = (−2, 4).
a + b = (3 − 2, −1 + 4) = (1, 3).
Find the magnitude of (6, −8).
√(6² + 8²) = √100 = 10.
What is −2a when a = (3, −1)?
(−6, 2): twice as long as a and in the opposite direction.
Calculate a · b for a = (2, 3) and b = (4, −1).
2 × 4 + 3 × (−1) = 8 − 3 = 5.
How can you tell from the dot product that two vectors are perpendicular?
Their dot product is zero, because cos 90° = 0.
When are two vectors parallel (collinear)?
When one is a scalar multiple of the other, e.g. (2, −3) and (−6, 9) = −3 × (2, −3).
For teachers: expected results and how the model works
Grid from −8 to 8; tap the grid to move the tip of the selected vector (integer components). a + b is shown with the triangle law (b drawn from the tip of a, the faint dashed copy of b from O shows the parallelogram); a − b is drawn from the tip of b to the tip of a; k × a uses k from −3 to 3. The angle uses cos θ = a · b ÷ (|a||b|) and is always between 0° and 180°. “Make b perpendicular to a” sets b = (−a₂, a₁), halved if it leaves the grid. Default a = (4, 2), b = (1, 5): a + b = (5, 7), |a + b| = 8.60, a · b = 14, θ = 52.1°.
Exam practice Exam set
National-exam style questions on this simulation (17 marks). Try each one on paper first, then open the marking scheme.
Given a = (3, 4) and b = (−2, 1), find (a) a + b, (b) |a|, (c) a · b, (d) the angle between a and b. (7 marks)
Marking scheme: (a) (1, 5) (1). (b) √(9 + 16) = 5 (1). (c) 3 × (−2) + 4 × 1 = −2 (2). (d) cos θ = −2 ÷ (5 × √5) = −0.179 (2); θ = 100.3° (1).
Find k if the vectors (k, 2) and (3, −6) are perpendicular. (3 marks)
Marking scheme: (k, 2) · (3, −6) = 3k − 12 (1); 3k − 12 = 0 (1); k = 4 (1).
Show that the points A(1, 2), B(3, 5) and C(7, 11) are collinear. (3 marks)
Marking scheme: AB = (2, 3) (1); AC = (6, 9) = 3AB (1); AC is a multiple of AB and they share point A, so A, B and C are collinear (1).
A boat heads across a river at 4 m/s while the current flows at 3 m/s at right angles. Find the resultant speed and the angle it makes with the boat’s heading. (4 marks)
Marking scheme: Resultant = √(4² + 3²) = 5 m/s (2); angle = tan⁻¹(3 ÷ 4) = 36.9° (2).
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Vectors: addition, subtraction, scalar multiples and the dot product
Aim: To represent vectors as column vectors and arrows, to add and subtract vectors and multiply them by a scalar, to calculate magnitudes, and to use the dot product to find the angle between two vectors and test for perpendicular and parallel vectors.
Apparatus (real laboratory): squared paper, ruler, protractor.
What to do (with the simulation on www.donatstem.co.rw)
- Choose a + b. Tap the grid to set a = (4, 2) and b = (1, 5). Check a + b by adding the components. Record.
- Calculate |a|, |b| and |a + b|. Is |a + b| equal to |a| + |b|? When could it be?
- Choose a − b. Where does the arrow a − b start and end?
- Choose k × a. Try k = 2, −1 and 0.5. What do you notice about the direction?
- Choose a · b and the angle. Tap Make b perpendicular to a. What is a · b?
- Use New vectors several times. Predict whether θ is acute or obtuse from the sign of a · b, then check.
Results
| # | a | b | Operation | Result | |Result| | a · b | θ (°) |
|---|
| 1 | | | | | | | |
| 2 | | | | | | | |
| 3 | | | | | | | |
| 4 | | | | | | | |
| 5 | | | | | | | |
| 6 | | | | | | | |
| 7 | | | | | | | |
| 8 | | | | | | | |
| 9 | | | | | | | |
| 10 | | | | | | | |
Graph
No graph: draw each pair of vectors and its result on squared paper.
Questions
- Find a + b when a = (3, −1) and b = (−2, 4). [2]
- Find the magnitude of (6, −8). [2]
- What is −2a when a = (3, −1)? [2]
- Calculate a · b for a = (2, 3) and b = (4, −1). [2]
- How can you tell from the dot product that two vectors are perpendicular? [2]
- When are two vectors parallel (collinear)? [2]
- Given a = (3, 4) and b = (−2, 1), find (a) a + b, (b) |a|, (c) a · b, (d) the angle between a and b. [7]
- Find k if the vectors (k, 2) and (3, −6) are perpendicular. [3]
- Show that the points A(1, 2), B(3, 5) and C(7, 11) are collinear. [3]
- A boat heads across a river at 4 m/s while the current flows at 3 m/s at right angles. Find the resultant speed and the angle it makes with the boat’s heading. [4]
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Marking scheme: Vectors: addition, subtraction, scalar multiples and the dot product
Expected results
Grid from −8 to 8; tap the grid to move the tip of the selected vector (integer components). a + b is shown with the triangle law (b drawn from the tip of a, the faint dashed copy of b from O shows the parallelogram); a − b is drawn from the tip of b to the tip of a; k × a uses k from −3 to 3. The angle uses cos θ = a · b ÷ (|a||b|) and is always between 0° and 180°. “Make b perpendicular to a” sets b = (−a₂, a₁), halved if it leaves the grid. Default a = (4, 2), b = (1, 5): a + b = (5, 7), |a + b| = 8.60, a · b = 14, θ = 52.1°.
Answers
- Find a + b when a = (3, −1) and b = (−2, 4). [2]
a + b = (3 − 2, −1 + 4) = (1, 3).
- Find the magnitude of (6, −8). [2]
√(6² + 8²) = √100 = 10.
- What is −2a when a = (3, −1)? [2]
(−6, 2): twice as long as a and in the opposite direction.
- Calculate a · b for a = (2, 3) and b = (4, −1). [2]
2 × 4 + 3 × (−1) = 8 − 3 = 5.
- How can you tell from the dot product that two vectors are perpendicular? [2]
Their dot product is zero, because cos 90° = 0.
- When are two vectors parallel (collinear)? [2]
When one is a scalar multiple of the other, e.g. (2, −3) and (−6, 9) = −3 × (2, −3).
- Given a = (3, 4) and b = (−2, 1), find (a) a + b, (b) |a|, (c) a · b, (d) the angle between a and b. [7]
(a) (1, 5) (1). (b) √(9 + 16) = 5 (1). (c) 3 × (−2) + 4 × 1 = −2 (2). (d) cos θ = −2 ÷ (5 × √5) = −0.179 (2); θ = 100.3° (1).
- Find k if the vectors (k, 2) and (3, −6) are perpendicular. [3]
(k, 2) · (3, −6) = 3k − 12 (1); 3k − 12 = 0 (1); k = 4 (1).
- Show that the points A(1, 2), B(3, 5) and C(7, 11) are collinear. [3]
AB = (2, 3) (1); AC = (6, 9) = 3AB (1); AC is a multiple of AB and they share point A, so A, B and C are collinear (1).
- A boat heads across a river at 4 m/s while the current flows at 3 m/s at right angles. Find the resultant speed and the angle it makes with the boat’s heading. [4]
Resultant = √(4² + 3²) = 5 m/s (2); angle = tan⁻¹(3 ÷ 4) = 36.9° (2).