Addition and subtraction of vectors
Walk 3 km east and then 4 km north. You end up 5 km away from the start in a north-east direction. Vector addition joins the two steps into a single displacement.
By the triangle law, if a is drawn from A to B and b from B to C, then a + b is the vector from A to C. In components, add the corresponding components. Subtraction a − b means a + (−b).
Formula: AB + BC = AC, |a + b| ≤ |a| + |b|
Rule 1: Add component by component — Match i with i, j with j, k with k.
- Easy: (2i + 3j) + (i − j) = 3i + 2j.
- Medium: (i + 2j − k) − (3i − j + 2k) = −2i + 3j − 3k.
- Exam-type: For a = 2i + j − 2k and b = i − j + 2k, a + b = 3i and |a + b| = 3.
Rule 2: Triangle inequality — A direct route is never longer than a detour.
- Easy: For a = 3i and b = 4j, |a + b| = 5 and |a| + |b| = 7.
- Medium: For a = 2i + j and b = −2i + 2j, |a + b| = 3.
- Exam-type: AB + BC + CA = 0 for any triangle ABC.
Add corresponding components. The magnitude of a sum is not the sum of the magnitudes.
Writing |a + b| = |a| + |b|, which is true only when the vectors point the same way.
Tip to tail.
Practice questions
- 1Find (2i + 3j) + (i − j).
- 2Find (i + 2j − k) − (3i − j + 2k).
- 3Find |(2i + j) + (−2i + 2j)|.
- 4Find a − b for a = 2i + j − 2k and b = i − j + 2k.
- 5Find |a + b| for a = 2i + j − 2k and b = i − j + 2k.
- 6Find 2a + 3b for a = i + j and b = i − j.
- 7Verify |a + b| ≤ |a| + |b| for a = 3i and b = 4j.
- 8What is AB + BC + CA for a triangle ABC?
Show answers (try them yourself first)
- 13i + 2j
- 2−2i + 3j − 3k
- 3|3j| = 3
- 4i + 2j − 4k
- 5a + b = 3i, so the magnitude is 3
- 62i + 2j + 3i − 3j = 5i − j
- 7|a + b| = 5 and |a| + |b| = 7, so 5 ≤ 7
- 8The zero vector
Multiplication of a vector by a scalar
Doubling a recipe doubles every quantity but does not change the taste. Multiplying a vector by a number stretches or shrinks it, and a negative number also turns it around.
For a number λ, the vector λa has magnitude |λ| |a|. Its direction is the same as a if λ > 0 and opposite if λ < 0. The unit vector along a is a / |a|, and a vector of magnitude m along a is m a / |a|. Two vectors are collinear if one is a scalar multiple of the other.
Formula: |λa| = |λ| |a|, vector of magnitude m along a = m a / |a|
Rule 1: Scale and normalise — Multiply every component by the number.
- Easy: 3(2i − j) = 6i − 3j.
- Medium: The unit vector along 6i + 8j is (3i + 4j)/5.
- Exam-type: A vector of magnitude 14 along 2i + 3j + 6k is 2(2i + 3j + 6k) = 4i + 6j + 12k.
Rule 2: Collinear — Check whether the components are proportional.
- Easy: 2i + 4j = 2(i + 2j), so they are collinear.
- Medium: For A(1, 1), B(2, 3) and C(3, 5), AB = i + 2j and BC = i + 2j, so A, B, C are collinear.
- Exam-type: 2i + λj is parallel to 4i + 6j if λ = 3.
A negative multiple points the opposite way. Components are multiplied one by one.
Multiplying only the first component by the scalar.
Scale every component, and use the length for a unit vector.
Practice questions
- 1Find 3(2i − j).
- 2Find −3(i − 2j + k).
- 3Find |3a| if a = 2i + 2j + k.
- 4Find the unit vector along 6i + 8j.
- 5Find the vector of magnitude 14 along 2i + 3j + 6k.
- 6Find the vector of magnitude 5 in the direction opposite to 3i + 4j.
- 7Are 2i + 4j and i + 2j collinear?
- 8Find λ so that 2i + λj is parallel to 4i + 6j.
Show answers (try them yourself first)
- 16i − 3j
- 2−3i + 6j − 3k
- 33 × 3 = 9
- 4(3i + 4j)/5
- 54i + 6j + 12k
- 6−(3i + 4j) = −3i − 4j
- 7Yes, since the first is twice the second
- 82/4 = λ/6, so λ = 3