Vectors in Two Dimensions
Grade 11
1. Revision: Vectors and Scalars in One Dimension
Revision from Grade 10
Everything in this section was first taught in Grade 10. Grade 11 builds directly on it, so work through it again before moving on to two dimensions.
A physical quantity is anything in physics that can be measured, such as time, mass, weight, force and charge. Some quantities are fully described by a number and a unit; others also need a direction.
Definition: scalar
A scalar is a physical quantity that has magnitude only.
Definition: vector
A vector is a physical quantity that has both magnitude and direction.
Scalars
- mass (kg)
- time (s)
- distance (m)
- speed (m·s⁻¹)
- charge (C)
- energy (J)
Vectors
- force (N)
- weight (N)
- displacement (m)
- velocity (m·s⁻¹)
- acceleration (m·s⁻²)
A mass of 5 kg is complete as it stands, but a force of 5 N is not: it matters whether you push the box to the left or to the right. Weight is a force, so it is a vector that always points downwards, while mass is a scalar.
Writing vectors. A symbol with an arrow above it, F→, represents the force vector: its magnitude and its direction. The same letter without the arrow, F, represents only the magnitude of the force, for example F = 20 N.
Drawing vectors. A vector is drawn as an arrow. The length of the arrow, drawn to a chosen scale, shows the magnitude, and the arrowhead shows the direction. The start of the arrow is its tail and the end with the arrowhead is its head.
Two force vectors drawn to scale: 40 N to the right and 20 N to the left. The first arrow is twice as long as the second because its force is twice as large.
Directions in one dimension. Along a straight line there are only two directions. Choose one as positive (for example to the right or east); the opposite direction is then negative. A force of −20 N, with right as positive, is 20 N to the left.
Adding vectors in one dimension
Equal vectors have the same magnitude and the same direction. The negative of a vector has the same magnitude but the opposite direction: if A is 30 N to the right, −A is 30 N to the left.
A and B are equal vectors (30 N to the right). C is the negative of A (30 N to the left). D points the same way as A but is larger, so it is not equal to A.
Definition: resultant vector
The resultant vector is the single vector that has the same effect as all the original vectors acting together.
The resultant of force vectors is also called the net force. It can be found by drawing or by calculation.
The tail-to-head method
- Choose a scale. Draw a start line.
- On the top line, draw the vectors in the direction with the larger total one after the other, tail to head, starting at the start line.
- On the line underneath, draw the vectors in the opposite direction tail to head, starting at the head of the last vector on the top line and pointing back.
- Draw the resultant on the bottom line from the start line to the head of the last vector there, so the two arrowheads meet. Measure its length and use the scale to find its magnitude; it points the same way as the top line.
Forces of 15 N east, 40 N west and 10 N east drawn tail to head, with east to the right. F₂ points the way with the larger total, so it goes on the top line. On the line underneath, F₁ and F₃ follow tail to head, starting at the head of F₂. R runs along the bottom line from the start line to the head of F₃. R is 3 squares long, which is 15 N west.
Finding the resultant by calculation
- Choose a positive direction.
- Give each vector a sign: + in the positive direction and − in the opposite direction.
- Add the signed values.
- Give the magnitude of the answer and turn its sign back into a direction.
Forces in the same direction add up to a larger resultant; forces in opposite directions partly or completely cancel. For two forces along a line, the largest possible resultant is their sum and the smallest is their difference.
Subtracting vectors. To subtract a vector, add its negative: A − B = A + (−B). This works for up to four force vectors in the same way, one at a time.
Three horizontal forces on a crate. With east as positive, R = (+30 N) + (+12 N) + (−50 N) = −8 N, so the resultant is 8 N west.
Worked Examples
- 150 kg is a mass: magnitude only, so a scalar.
- 220 N downwards is a force with a direction: a vector.
- 33 s is a time: a scalar.
- 412 m east is a displacement with a direction: a vector.
- 1length = 45 N ÷ 10 N per cm
- 1Take east as positive: R = (+15 N) + (−40 N) + (+10 N) = −15 N, so R = 15 N west.
- 1Take right as positive: A = +25 N and B = −10 N.
- 2A − B = A + (−B) = (+25 N) + (+10 N) = +35 N
2. Resultant of Perpendicular Vectors
In Grade 10 all the vectors acted along one straight line. In two dimensions they can act in any direction in a flat plane, so we work on a Cartesian plane: a horizontal x-axis and a vertical y-axis that cross at right angles. East (or to the right) is usually the positive x-direction and north (or upwards) the positive y-direction.
Definition: resultant vector
The resultant vector is the single vector that has the same effect as all the original vectors acting together.
The method works for force vectors and for displacement vectors, and for up to four vectors at a time.
Step 1: add the co-linear vectors on each axis
Vectors along the same line are co-linear. Add the co-linear horizontal vectors, with signs, to get the net horizontal vector Rₓ. Add the co-linear vertical vectors to get the net vertical vector Rᵧ. This is the Grade 10 method, used once for each axis.
Four forces act on one point. Horizontally: Rₓ = (+20 N) + (−8 N) = +12 N, so Rₓ = 12 N east. Vertically: Rᵧ = (+14 N) + (−5 N) = +9 N, so Rᵧ = 9 N north.
Step 2: sketch Rₓ and Rᵧ and the resultant
Sketch Rₓ and Rᵧ on the Cartesian plane, then sketch the resultant R in one of two ways:
- Tail-to-head method: draw Rₓ, then draw Rᵧ with its tail at the head of Rₓ. R runs from the tail of Rₓ to the head of Rᵧ.
- Tail-to-tail (parallelogram) method: draw Rₓ and Rᵧ from the same point (tail to tail). Complete the parallelogram with dashed lines; because Rₓ and Rᵧ are perpendicular, it is a rectangle. R is the diagonal that starts at the common tail.
Tail to head: Rᵧ starts at the head of Rₓ, and R closes the right-angled triangle. θ is the angle between R and the x-axis.
Tail to tail: Rₓ and Rᵧ start at the same point. The rectangle is completed with dashed lines, and R is the diagonal from the common tail.
Step 3: the magnitude, with the theorem of Pythagoras
Rₓ, Rᵧ and R form a right-angled triangle with R as the hypotenuse, so the theorem of Pythagoras gives the magnitude of the resultant:
R² = Rₓ² + Rᵧ²
Step 4: the direction, with a trigonometric ratio
The angle θ between R and the x-axis follows from the tangent ratio. Use the magnitudes of the components, then describe the direction in words from the sketch.
tan θ = RᵧRₓ
Give the direction as an angle from a named direction, for example "36,87° north of east" or "36,87° above the positive x-axis". A direction can also be given as a compass bearing, measured clockwise from north: 36,87° north of east is a bearing of 53,13°.
Common mistakes
- Adding the magnitudes: 12 N east and 9 N north do not give 21 N, because they do not act along one line.
- Forgetting the square root: R² = 225 N², so R = 15 N, not 225 N.
- Turning the ratio upside down: tan θ = RₓRᵧ gives the angle from the y-axis, not from the x-axis.
- Giving the size without the direction: a resultant vector needs both.
Finding the resultant by drawing. Choose a scale, draw the vectors tail to head with a ruler and protractor, and draw R from the tail of the first vector to the head of the last. Measure the length of R and convert it with the scale; measure θ with the protractor. This graphical tail-to-head method works for up to four vectors in any directions. The calculation (component method) gives the same answer without a scale drawing.
Definition: closed vector diagram
A closed vector diagram is a vector diagram in which the vectors are drawn tail to head and the head of the last vector ends at the tail of the first vector, so that the resultant of the vectors is zero.
If the vectors drawn tail to head end exactly where the first one started, there is no gap for a resultant to fill: the resultant is zero. For forces, a closed vector diagram means the net force on the object is zero. Three forces that keep an object in equilibrium always form a closed triangle, and any one of them is equal in size and opposite in direction to the resultant of the other two.
F₁ = 12 N east, F₂ = 9 N north and F₃ = 15 N at 36,87° south of west, drawn tail to head. The head of F₃ ends at the tail of F₁: a closed vector diagram, so the resultant is zero.
Practical: forces on a force board
Hang masses from three strings tied at a knot, with two strings running over pulleys at the edges of a force board. When the knot stays still, draw the three forces (weights) to scale in their directions. Drawn tail to head, they form a closed triangle, which shows that the resultant of the three non-linear forces is zero.
How to plan, record and write up a practical: see the scientific investigation skills guide.
Worked Examples
- 1Take east and north as positive. Horizontally: Rₓ = (+20 N) + (−8 N) = +12 N, so Rₓ = 12 N east.
- 2Vertically: Rᵧ = (+14 N) + (−5 N) = +9 N, so Rᵧ = 9 N north.
- 3R² = Rₓ² + Rᵧ² = (12)² + (9)² = 225
- 4R = 15 N
- 5tan θ = 912, so θ = 36,87°
- 1The two displacements are perpendicular: Δy = 1,2 km north and Δx = 0,9 km east.
- 2Δx² + Δy² = R², so R² = (0,9)² + (1,2)² = 2,25
- 3R = 1,5 km
- 4tan θ = 1,20,9, so θ = 53,13° (from east, towards north)
3. Resolving a Vector into Components
Resolving works in the opposite direction to finding a resultant. Any vector can be replaced by two perpendicular vectors, a horizontal x-component and a vertical y-component, that together have exactly the same effect as the original vector. The vector is the resultant of its components.
How to resolve a vector
- Draw a sketch of the vector on the Cartesian plane, starting at the origin. Show its magnitude and the angle θ between the vector and the x-axis.
- Drop dashed lines from the head of the vector to each axis to show the components.
- Calculate the components with the formulas below.
- Give each component a sign or a direction from the sketch: a component pointing left (west) or down (south) is negative.
Rₓ = R cos θ
Rᵧ = R sin θ
A force of 80 N at 30° to the positive x-axis. Its components are Fₓ = 80 N × cos 30° = 69,28 N and Fᵧ = 80 N × sin 30° = 40 N.
Check where the angle is measured from
R cos θ gives the x-component only when θ is measured from the x-axis: the x-component is the side adjacent to θ. If the angle is given from the vertical (the y-axis), the x-component is opposite that angle, so it is R sin of that angle, and the y-component is R cos of that angle. Always sketch the vector first.
Force and displacement. The same method resolves any vector. A suitcase pulled with a force along a handle that slopes upwards moves because of the horizontal component of the pull, while the vertical component lifts some of its weight. A displacement of 15 km at 40° north of west has a westward component of 15 cos 40° = 11,49 km and a northward component of 15 sin 40° = 9,64 km.
A displacement of 15 km at 40° north of west. The angle is measured from the west side of the x-axis, so the x-component is −11,49 km (west) and the y-component is +9,64 km (north).
The component method for up to four vectors
- Resolve every vector that is not along an axis into its x- and y-components.
- Add all the x-components, with signs, to get Rₓ. Add all the y-components to get Rᵧ.
- Sketch Rₓ and Rᵧ and find R with R² = Rₓ² + Rᵧ².
- Find the direction with tan θ = RᵧRₓ and describe it from the sketch.
- If Rₓ = 0 and Rᵧ = 0, the vectors form a closed vector diagram and the resultant is zero.
Worked Examples
- 1Sketch the force from the origin at 30° above the positive x-axis.
- 2Fₓ = F cos θ = 80 × cos 30° = 69,28 N
- 3Fᵧ = F sin θ = 80 × sin 30° = 40 N
- 1Resolve the 50 N force: x-component = 50 cos 60° = 25 N east; y-component = 50 sin 60° = 43,3 N north.
- 2Rₓ = (+30) + (+25) + (−20) = 35 N, so Rₓ = 35 N east
- 3Rᵧ = (+43,3) + (−25) = 18,3 N, so Rᵧ = 18,3 N north
- 4R² = (35)² + (18,3)², so R = 39,5 N
- 5tan θ = 18,335, so θ = 27,6°