Vectors in 2D and 3D
Vectors describe quantities that possess both magnitude and direction in two-dimensional and three-dimensional space. In A-Level Maths, column vectors and i, j, k unit vectors are used to find distances, unit vectors, position vectors, and geometric relationships between points and lines.
- Unit Vector
- A vector that has a magnitude of exactly 1. Found by dividing a vector by its magnitude.
- Position Vector
- A vector that gives the position of a point relative to a fixed origin, O.
- Vector Equation of a Line
- Expressed as r = a + tb, where 'a' is a position vector on the line, 'b' is a direction vector, and 't' is a scalar parameter.
Vectors in 2D and 3D: 3D Molecular, Lattice, DNA & Vector Explorer
Drag directly on the 3D model to rotate it through any angle, or switch tabs to inspect VSEPR bond angles, crystal lattices, DNA base pairing, and 3D space diagonals.
Position Vector r = (4i + 3j + 5k)
- 1. Base Diagonal √(x² + y²):
- √(4² + 3²) = 5.00
- 2. 3D Magnitude |r| = √(x² + y² + z²):
- √50 = 7.07
- 3. Angle with Base tan⁻¹(z / base):
- 45.0°
To find the angle between the space diagonal OP and the horizontal base plane, form the shaded vertical right-angled triangle with adjacent = √(x² + y²) and opposite = z.
Key Concepts & Exam Definitions (5)
- Unit Vector
- A vector that has a magnitude of exactly 1. Found by dividing a vector by its magnitude.
- Position Vector
- A vector that gives the position of a point relative to a fixed origin, O.
- Vector Equation of a Line
- Expressed as r = a + tb, where 'a' is a position vector on the line, 'b' is a direction vector, and 't' is a scalar parameter.
- Magnitude
- The length or size of a vector, denoted by |a|.
- Collinear Points
- Points that lie on the same straight line. Proved by showing vectors between the points are parallel and share a common point.
Quick-Fire Flashcards (5)
Click any card to check the answer, then mark it as Known to track your progress.
What are the standard unit vectors in 3D?
How do you find the vector AB given position vectors a and b?
What formula finds the magnitude of vector xi + yj + zk?
How do you prove two vectors are parallel?
What does r represent in the line equation r = a + tb?
Exam-Style Questions & Mark Schemes (3)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
Find the magnitude of the vector 3i - 4j + 12k.
Click “Check Mark Scheme” to compare your answer, √(3² + (-4)² + 12²) = √(9 + 16 + 144) = √169 = 13.
- Q2
If point A has position vector 2i + j and point B has 5i - 3j, find the vector AB.
Click “Check Mark Scheme” to compare your answer, AB = b - a = (5i - 3j) - (2i + j) = 3i - 4j.
- Q3
Find the unit vector in the direction of 3i + 4j.
Click “Check Mark Scheme” to compare your answer, Magnitude is √(3²+4²) = 5. Unit vector is (3/5)i + (4/5)j.
Common Exam Questions (3)
What is the difference between a scalar and a vector?
A scalar quantity has magnitude (size) only, such as mass, speed, or temperature. A vector quantity has both magnitude and direction, such as displacement, velocity, acceleration, or force.
How do you find the magnitude of a 3D vector?
Apply Pythagoras' theorem in three dimensions by squaring each component, adding them together, and taking the square root: |v| = √(x² + y² + z²).
What is a position vector?
A position vector gives the displacement of a point P relative to the fixed origin O, written as OP or r, whose components match the coordinates of point P.
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