Vectors — Edexcel A-Level Mathematics
Test yourself on Vectors with PEARSON EDEXCEL A-Level practice questions.
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Vectors explained
A vector has magnitude and direction and can be written in component form.
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In two dimensions a = a₁i + a₂j, and in three dimensions a = a₁i + a₂j + a₃k, where i, j and k are unit vectors along the coordinate axes. You add and subtract vectors component by component, multiply by a scalar, and find the vector between two points by subtracting position vectors. For example, if A(1, 2, 3) and B(4, 0, 5), then AB = (4 − 1)i + (0 − 2)j + (5 − 3)k = 3i − 2j + 2k. Vectors model displacement, velocity and force in both dimensions, and the same component rules apply in 2-D and 3-D.
10.2 Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form.
The magnitude of a vector is its length. In two dimensions, if a = xi + yj then |a| = √(x² + y²); in three dimensions, if a = xi + yj + zk then |a| = √(x² + y² + z²). Direction in 2-D is the angle θ measured from the positive x-axis, found using tan θ = y ÷ x with the correct quadrant. Converting from magnitude/direction form back to components uses x = r cos θ and y = r sin θ, where r = |a|. For example, a vector of magnitude 5 at 30° to the positive x-axis has components x = 5 cos 30° = (5√3) ÷ 2 and y = 5 sin 30° = 5 ÷ 2. In three dimensions, direction is described by direction cosines or by angles with the axes rather than a single bearing.
10.3 Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations.
Vectors carry both magnitude and direction, so they can be represented by directed line segments and by column vectors such as (3, -2). Diagrammatically, place the tail of the second vector at the head of the first; the resultant runs from the first tail to the final head, forming a triangle or parallelogram. Algebraically, add componentwise: (a, b) + (c, d) = (a + c, b + d). Multiplying by scalar k scales every component, so k(a, b) = (ka, kb); its magnitude scales by |k| and its direction reverses when k is negative. Geometrically, addition gives the net displacement, while scalar multiplication produces parallel vectors, underpinning collinearity and ratio proofs.
10.4 Understand and use position vectors; calculate the distance between two points represented by position vectors.
A position vector fixes a point relative to a chosen origin, usually written as a column vector from O to the point. If A has position vector a and B has position vector b, then the vector from A to B is b - a, found by subtracting coordinates componentwise. The distance AB is the magnitude of this displacement: |b - a| = √((x₂ - x₁)² + (y₂ - y₁)²) in two dimensions, extending to √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²) in three dimensions. This links position vectors directly to Pythagoras and to coordinate geometry, and it supports finding lengths, midpoints and unit vectors.
10.5 Use vectors to solve problems in pure mathematics and in context (including forces).
Vectors model quantities with magnitude and direction, so they solve pure geometry and real contexts. In pure mathematics, express lines and points with position vectors, prove collinearity by showing one vector is a scalar multiple of another, and find ratios using section formulae. In context, resolve forces into components, add them to find a resultant, and interpret equilibrium when the resultant is zero. For example, forces of 3 N east and 4 N north give a resultant of 5 N on a bearing found by trigonometry. Always define a positive direction, keep units consistent, and interpret the final vector in the setting.
Your focus
- Express and manipulate vectors in two and three dimensions using component notation.
- Find displacement vectors between points and use them to solve geometric problems.
- Apply vector methods to model and solve problems involving ratios, midpoints and parallel vectors.
Show all 15 objectives
- Calculate the magnitude of a vector in two or three dimensions from its components.
- Determine the direction of a two-dimensional vector and convert between component and magnitude/direction forms.
- Describe the direction of a three-dimensional vector using direction cosines or angles with the coordinate axes.
- Add two or more vectors diagrammatically and algebraically with correct components.
- Multiply a vector by a scalar and describe the effect on magnitude and direction.
- Interpret a resultant vector geometrically in a displacement or force context.
- Write and interpret position vectors for points in two and three dimensions.
- Calculate the displacement vector and distance between two points from their position vectors.
- Apply position vectors to solve a simple geometric problem accurately.
- Model a contextual or geometric problem using vectors with a defined positive direction.
- Find resultants and solve equilibrium problems involving forces.
- Use vector methods to prove collinearity and interpret the result in context.
Vectors exam tips
Marking Points
- Writes vectors in component form using i, j and, in three dimensions, k, and interprets each component as a displacement along an axis.
- Adds, subtracts and multiplies vectors by scalars component by component, including finding AB as b − a from position vectors.
- Uses vectors to model and solve geometric or physical problems in two and three dimensions, such as finding a point dividing a line in a given ratio.
- Solves problems involving parallel vectors by equating components or using scalar multiples, and recognises when vectors are equal or parallel.
- Calculates the magnitude of a 2-D or 3-D vector using the square root of the sum of squared components, retaining exact surd form where appropriate.
- Finds the direction of a 2-D vector using tan θ = y ÷ x and selects the correct quadrant from the signs of the components.
- Converts from magnitude/direction form to component form using x = r cos θ and y = r sin θ, and from component form to magnitude/direction form.
- Uses direction cosines or angles with the coordinate axes to describe the direction of a three-dimensional vector.
- Draws directed line segments with correct relative directions and joins them head-to-tail to form the resultant.
- Adds vectors componentwise, for example (2, 5) + (-1, 3) = (1, 8), keeping components aligned.
- Multiplies a vector by a scalar, for example 3(2, -1) = (6, -3), and states that the result is parallel to the original.
- Interprets the resultant geometrically as a net displacement or combined effect and interprets a negative scalar as reversing direction.
- Uses vector addition and scalar multiplication together in a simple proof, such as showing three points are collinear via a common scalar multiple.
- Writes the position vector of a point as a column vector from the origin with correct components.
- Finds the displacement vector between two points by subtracting position vectors in the correct order, b - a.
- Calculates the distance between two points using the magnitude formula, including the three-dimensional case.
- Uses a calculated distance or displacement to answer a geometric question, such as identifying an equal length or a right angle.
- Maintains correct signs when coordinates are negative, for example subtracting a negative coordinate increases the component.
- Models a context with vectors, choosing axes and stating a positive direction before resolving.
- Adds or subtracts vectors to find a resultant and interprets its magnitude and direction in context.
- Proves collinearity or finds a ratio by expressing a vector as a scalar multiple of another and identifying a shared point.
- Uses trigonometry and Pythagoras' theorem to find the magnitude and direction of a vector from its components.
- Applies equilibrium by setting the vector sum of forces to zero and solving for an unknown force.
Examiner Tips
- 💡Write vectors in a clear column or i, j, k form and keep components aligned to avoid arithmetic errors.
- 💡When a ratio or midpoint is involved, express the required point as a position vector using a parameter and then solve component equations.
- 💡Check whether the question is set in 2-D or 3-D before deciding how many components to include.
- 💡Sketch the vector in the correct quadrant before calculating the direction angle to avoid sign errors.
- 💡Keep exact values such as surds and fractions unless the question asks for a decimal, and give angles to the accuracy requested.
- 💡For 3-D vectors, state clearly whether you are giving direction cosines or angles with the axes, and show the component calculations.
- 💡Sketch the vectors before calculating so the expected direction of the resultant is clear.
- 💡Keep column vectors aligned vertically when adding or subtracting to avoid component errors.
- 💡Check a collinearity claim by confirming one vector is a scalar multiple of the other and that the vectors share a point.
- 💡Label the two points clearly before subtracting so the order of subtraction is unambiguous.
- 💡Leave the squared components exact until the final square root to reduce rounding error.
- 💡Check whether the question asks for a vector or a scalar distance, since the magnitude is a single non-negative number.
- 💡Draw a clear diagram with labelled directions and known values before forming equations.
- 💡State the vector equation you are using, then substitute values, so method marks are visible.
- 💡Interpret the final answer in the context, including magnitude, direction and appropriate units.
Common Mistakes
- Reversing the subtraction when finding AB, giving a − b instead of b − a; correct this by remembering that AB means from A to B, so subtract the position vector of A from that of B.
- Treating a three-dimensional vector as if it had only two components; correct this by including the k component whenever the problem is in 3-D.
- Adding corresponding components incorrectly by mixing i, j and k terms; correct this by keeping like components together in separate columns or brackets.
- Forgetting to square each component before adding when finding magnitude; correct this by applying |a| = √(x² + y²) or |a| = √(x² + y² + z²).
- Using the wrong quadrant for the direction angle; correct this by checking the signs of the components and adjusting the angle accordingly.
- Swapping sine and cosine when converting to components; correct this by using x = r cos θ for the horizontal component and y = r sin θ for the vertical component.
- Subtracting components in the wrong order when a vector is reversed; correct this by negating both components before adding.
- Adding magnitudes instead of adding components; correct this by resolving into components and adding componentwise.
- Treating scalar multiplication as changing direction for every scalar; correct this by noting only a negative scalar reverses direction, while magnitude scales by the absolute value of the scalar.
- Subtracting position vectors in the reverse order and obtaining the opposite displacement; correct this by using b - a for the vector from A to B.
- Forgetting to square each component before adding when finding magnitude; correct this by applying the full distance formula.
- Dropping the square root after summing squares; correct this by taking the square root to obtain the actual distance.
- Adding force magnitudes without resolving direction; correct this by resolving into perpendicular components and adding componentwise.
- Claiming collinearity from parallel vectors that do not share a point; correct this by checking a common point as well as a scalar multiple.
- Mixing units or directions mid-calculation; correct this by fixing units and a positive direction at the start and keeping them throughout.