Vectors in Two Dimensions: Notation, Magnitude and Arithmetic
A vector has both magnitude and direction, and in two dimensions it is written as a column with the horizontal component on top and the vertical component below. Vectors are added by adding the components, subtracted by subtracting them, and multiplied by a scalar by multiplying both components. The magnitude, written with vertical bars, is found by Pythagoras: the square root of the sum of the squares of the components. Two vectors are parallel exactly when one is a scalar multiple of the other, and the negative of a vector has the same magnitude but the opposite direction. These rules are the arithmetic that vector geometry proofs are built on.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Write every vector in column form before doing arithmetic, with the horizontal component on top, so that adding and subtracting is simply component by component.
- Add or subtract by combining the top entries and then the bottom entries separately. Do not mix them.
- Multiply by a scalar by multiplying BOTH components, including when the scalar is negative or fractional.
- Find a magnitude with Pythagoras: square each component, add, and take the square root. Leave the answer in surd form when the question asks for an exact value.
- Test whether two vectors are parallel by asking whether one column is a constant multiple of the other. Both components must use the SAME multiplier.
- Interpret the direction from the signs: a positive top entry means right and a negative one left; a positive bottom entry means up and a negative one down.
Worked example
a is the column vector 3 over 4 and b is the column vector -1 over 2. Find (a) a + 2b, (b) the magnitude of a, and (c) state whether the vector 6 over 8 is parallel to a.
- For 2b, multiply both components of b by 2: 2b is the column -2 over 4.
- Add component by component: a + 2b has top entry 3 + (-2) = 1 and bottom entry 4 + 4 = 8, so a + 2b is the column 1 over 8.
- For the magnitude of a, square the components: 3 squared = 9 and 4 squared = 16.
- Add and take the square root: the square root of 25 is 5, so the magnitude of a is 5.
- For the parallel test, compare 6 over 8 with 3 over 4: 6 = 2 times 3 and 8 = 2 times 4, so the same multiplier 2 works for both components.
- Conclusion: the vector 6 over 8 equals 2a, so it is parallel to a and twice its length.
Practice questions
Try each question, then tap to reveal the answer.
Q1Add the column vectors 2 over 5 and 4 over -3.Show answer
Answer: The column 6 over 2.
Q2Find the magnitude of the column vector 5 over 12.Show answer
Answer: The square root of (25 + 144) = the square root of 169 = 13.
Q3Multiply the column vector -3 over 2 by the scalar 4.Show answer
Answer: The column -12 over 8.
Q4Is the column 4 over 6 parallel to the column 2 over 3?Show answer
Answer: Yes, because 4 over 6 is exactly 2 times 2 over 3, the same multiplier for both components.
Q5Is the column 4 over 7 parallel to the column 2 over 3?Show answer
Answer: No. The top entries need a multiplier of 2 but the bottom entries would need 7/3, and the multiplier must be the same for both.
Q6What is the relationship between a vector and its negative?Show answer
Answer: They have the same magnitude but point in exactly opposite directions.
Q7Find the magnitude of the column vector 1 over 1, leaving your answer in surd form.Show answer
Answer: The square root of 2.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
p is the column vector 4 over -1 and q is the column vector -2 over 5. (a) Find 3p - q as a column vector. (b) Find the magnitude of 3p - q, giving your answer to 2 decimal places.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 5 available
The vector AB is the column 6 over 9 and the vector CD is the column k over 6. Given that AB and CD are parallel, find the value of k and state the ratio of their magnitudes.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
Free printable worksheet
Want more practice on paper? Download the vectors in two dimensions: notation, magnitude and arithmetic worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Next topics
Not quite what you needed?
Tell us what is missing on vectors in two dimensions: notation, magnitude and arithmetic, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.