Vectors: notation, magnitude and unit vectors Cambridge IGCSE Additional Mathematics revision
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In plain words
A vector is a movement: so far across, so far up. It has a size and a direction, and that makes it the right tool for displacement, velocity and force.
7 things to know
- A vector can be written as a column, or as ai + bj, where i is one unit across and j is one unit up.
- The magnitude of ai + bj is √(a² + b²). It is written |a|.
- Vectors are added and subtracted by components. Multiplying by a number multiplies both components.
- The position vector of a point A is the vector from the origin O to A. The vector from A to B is the position vector of B minus the position vector of A.
- A unit vector has magnitude 1. The unit vector in the direction of a is a ÷ |a|.
- Two vectors are parallel if one is a multiple of the other.
- If two vectors are equal, their i components are equal and their j components are equal. That gives two equations.
Worked example
a = 3i − 4j. Find |a|, the unit vector in the direction of a, and a vector of magnitude 15 in the same direction.
- |a| = √(3² + (−4)²) = √25 = 5.
- Unit vector = (3i − 4j) ÷ 5 = (3/5)i − (4/5)j.
- A vector of magnitude 15 is 15 times the unit vector: 9i − 12j.
Worked example
The position vectors of A and B are 2i + j and 8i + 9j. Find the vector from A to B and its magnitude.
- Vector AB = (8i + 9j) − (2i + j) = 6i + 8j.
- |AB| = √(36 + 64) = 10.
Tips and tricks
- From A to B is "B minus A": end minus start. The other way round gives the opposite direction.
- Underline vectors when writing by hand, or put an arrow over two letters. Correct notation is expected.
It lands in your notebook with its questions as flashcards.
Vectors: notation, magnitude and unit vectors: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is the magnitude of 6i + 8j?
√(36 + 64).
What is a unit vector?
Divide a vector by its magnitude to get one.
A has position vector 2i + 5j and B has position vector 6i + 2j. What is the vector from A to B?
B minus A.
Which vector is parallel to 2i + 3j?
It is 3 times 2i + 3j.
What is the unit vector in the direction of 3i + 4j?
Divide by the magnitude, 5.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 8 marks. Write your answers on paper, then check them.
Vectors: notation, magnitude and unit vectors
Cambridge IGCSE Additional Mathematics 0606 · 8 marks · papermunch.org
Name ______________________________ Date ______________
Find the magnitude of 5i + 12j.[2]
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13. √(25 + 144).
The position vectors of A and B are i + 4j and 7i − 4j. Find the vector from A to B, its magnitude, and the unit vector in its direction.[3]
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6i − 8j, with magnitude 10. The unit vector is 0.6i − 0.8j.
p = 2i + 3j and q = i − j. Find 2p − 3q.[3]
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i + 9j. (4i + 6j) − (3i − 3j).
Answers: Vectors: notation, magnitude and unit vectors
- 1. 13. √(25 + 144).
- 2. 6i − 8j, with magnitude 10. The unit vector is 0.6i − 0.8j.
- 3. i + 9j. (4i + 6j) − (3i − 3j).