Vectors Calculator
Calculate vector operations for 2D and 3D vectors: addition, subtraction, scalar multiplication, magnitude, unit vector, dot product, angle between vectors, cross product, and midpoint. Full step-by-step working for UK A-Level Maths.
3D vectors only
Midpoint of two position vectors
Vectors in A-Level Maths
A vector is a mathematical quantity that has both magnitude (size) and direction. Vectors appear throughout A-Level Maths and A-Level Further Maths, describing positions, displacements, velocities and forces. In 2D, vectors are written as column vectors or as xi + yj. In 3D, they include a z-component: xi + yj + zk.
Vector Addition and Subtraction
Add or subtract vectors component by component: (a₁, a₂, a₃) + (b₁, b₂, b₃) = (a₁+b₁, a₂+b₂, a₃+b₃). Vector addition is commutative (a+b = b+a) and associative. Geometrically, addition corresponds to placing vectors tip to tail.
Magnitude and Unit Vectors
The magnitude of a = (x, y) is |a| = √(x² + y²). For 3D: |a| = √(x² + y² + z²). The unit vector in the direction of a is â = a/|a|.
Unit vectors have magnitude 1 and are used to represent direction alone. The standard basis unit vectors are i = (1,0,0), j = (0,1,0), k = (0,0,1).
Dot Product (Scalar Product)
The dot product a·b = a₁b₁ + a₂b₂ + a₃b₃ gives a scalar. It also equals |a||b|cosθ, so:
- If a·b = 0 (and a, b ≠ 0): vectors are perpendicular (θ = 90°)
- If a·b > 0: acute angle between vectors
- If a·b < 0: obtuse angle between vectors
- θ = arccos(a·b / (|a||b|))
Cross Product (Vector Product)
The cross product a×b produces a vector perpendicular to both a and b. For 3D vectors: a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). The magnitude |a×b| = |a||b|sinθ equals the area of the parallelogram formed by a and b. Note: a×b = −(b×a).
Position Vectors and Midpoints
If A has position vector a and B has position vector b, the vector AB = b − a. The midpoint M of AB has position vector (a + b)/2. The section formula for point P dividing AB in ratio m:n gives position vector (nb + ma)/(m+n).
How the Vectors Calculator Works
This calculator uses the current UK grading system and educational standards to help students, parents, and teachers understand academic performance. UK education follows specific grading frameworks that differ between GCSEs, A-Levels, and university degrees.
Understanding how grades are calculated and what they mean for future progression is important for making informed decisions about subject choices, university applications, and career planning.
Key Information for 2025/26
GCSEs in England use the 9-1 grading scale, where 9 is the highest and 4 is a standard pass (equivalent to the old C grade). A-Levels use the A*-E scale with UCAS tariff points ranging from 56 (A*) to 16 (E). Universities typically require grades between AAA and CCC depending on the course and institution, with highly competitive courses often asking for A*A*A.
Example Calculation
A student achieving grades of A*, A, and B at A-Level would earn UCAS tariff points of 56 + 48 + 40 = 144 points total. This exceeds the typical entry requirement for most Russell Group universities (128 points or AAB equivalent) and would make the student competitive for many courses.
Source: Based on Ofqual and UCAS 2025/26 guidelines. Last updated March 2026.
Frequently Asked Questions
A vector is a mathematical object that has both magnitude (size) and direction, unlike a scalar which has only magnitude. Vectors are written as column vectors or using i, j, k unit vector notation. In 2D, a vector (3, 4) means 3 units in the x-direction and 4 units in the y-direction. In A-Level Maths, vectors are used to describe positions, displacements, velocities and forces.
The magnitude (or modulus) of a vector a = (x, y) in 2D is |a| = √(x² + y²). For a 3D vector a = (x, y, z), the magnitude is |a| = √(x² + y² + z²). This is derived from Pythagoras' theorem. For example, if a = (3, 4), then |a| = √(9 + 16) = √25 = 5.
The dot product (scalar product) of vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is a·b = a₁b₁ + a₂b₂ + a₃b₃. The result is a scalar (number), not a vector. The dot product also equals |a||b|cosθ, where θ is the angle between the vectors. If a·b = 0 and both vectors are non-zero, the vectors are perpendicular.
Use the formula cosθ = (a·b) / (|a||b|), then find θ = arccos((a·b) / (|a||b|)). First calculate the dot product a·b, then calculate the magnitudes |a| and |b|, then divide and apply arccos. For example, if a = (1, 0) and b = (0, 1), the dot product is 0, so cosθ = 0, giving θ = 90°. The angle is always between 0° and 180°.
The cross product of 3D vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is a vector perpendicular to both: a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). Its magnitude |a×b| = |a||b|sinθ. The cross product is not commutative: a×b = −(b×a).
If a×b = 0, the vectors are parallel. Cross products are used in A-Level Further Maths and physics.
A unit vector is a vector with magnitude 1. To find the unit vector in the direction of vector a, divide by its magnitude: â = a / |a|. For example, if a = (3, 4), |a| = 5, so the unit vector is (3/5, 4/5) = (0.6, 0.8).
Unit vectors are used to represent direction without specifying magnitude. The standard unit vectors in 3D are i = (1,0,0), j = (0,1,0) and k = (0,0,1).
If A has position vector a and B has position vector b, the midpoint M of AB has position vector m = (a + b)/2. Simply add the corresponding components and divide by 2. For example, if a = (2, 4) and b = (6, 8), then m = ((2+6)/2, (4+8)/2) = (4, 6). The midpoint theorem for vectors states that the vector to M equals the average of the vectors to A and B.