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What Are Basis Vectors Used For?

Knowledge of basis vectors is important for your basic understanding of vectors.

Four basis vectors and two vectors decomposed using the basis vectors

Theory

Important Concepts

  • A basis is a set of vectors you can combine in different ways to create all the other vectors in a plane.

  • A basis vector is a vector from this set.

Theory

Basis Vector

If you have two vectors x→ and y→ that are not parallel, you can use them as basis vectors. These basis vectors can be multiplied by any real number. This means that all other vectors in the plane can be written as a sum of these vectors multiplied by a number.

Example 1

Given the basis vectors (1,−1) and (3, 4), write (11, 10) using the basis vectors.

(11, 10) = k (1,−1) + l (3, 4) = (k,−k) + (3l, 4l)

(11, 10) = k (1,−1) + l (3, 4) = (k,−k) + (3l, 4l)

k + 3l = 11 k = 11 − 3l − k + 4l = 10 ⇓ − (11 − 3l) + 4l = 10 − 11 + 3l + 4l = 10 7l = 21| : 7 l = 3 k = 11 − 3l ⇓ k = 11 − 3 ⋅ 3 k = 2

k + 3l = 11 − k + 4l = 10 k = 11 − 3l −(11 − 3l) + 4l = 10 − 11 + 3l + 41 = 10 7l = 21| : 7 l = 3 k = 11 − 3 ⋅ 3 k = 2

This allows you to write

(11, 10) = 2 (1,−1) + 3 (3, 4) = (2,−2) + (9, 12)

(11, 10) = 2 (1,−1) + 3 (3, 4) = (2,−2) + (9, 12)

You can see that the vector (11, 10) can be written as a sum of the two basis vectors multiplied by the constants 2 and 3.

Example 2

Given the vectors

u→ = 2a→ −b→, v→ = −6a→ − 2b→,

find the angle between the vectors u→ and v→ when |a→| = 2, |b→| = 3 and a→ ⋅b→ = 3

The angle between u→ and v→ is

cos ⁡ α = u→ ⋅v→ |u→| ⋅|v→|

First, you need to find u→ ⋅v→:

u→ ⋅v→ = (2a→ −b→) ⋅(−6a→ − 2b→) = −12a→2 − 4a→ ⋅b→ + 6a→ ⋅b→ + 2b→2 = −12a→2 + 2a→ ⋅b→ + 2b→2 = −12 |a→|2 + 2a→ ⋅b→ + 2 |b→|2 = −12 ⋅ 22 + 2 ⋅ 3 + 2 ⋅ 32 = −48 + 6 + 18 = −24

You also need to calculate the length of u→ and v→. Remember that a→2 = |a→|2.

|u→|2 = (2a→ −b→) ⋅ (2a→ −b→) = 4 |a→|2 − 4a→ ⋅b→ + |b→|2 = 4 ⋅ 22 − 4 ⋅ 3 + 32 = 16 − 12 + 9 = 13 |u→| = |u→ | 2 = 13, |v→|2 = (−6a→ − 2b→) ⋅ (−6a→ − 2b→) = 36 |a→|2 − 24a→ ⋅b→ + 4 |b→|2 = 36 ⋅ 22 − 24 ⋅ 3 + 4 ⋅ 32 = 144 − 72 + 36 = 108 |v→| = |v→ | 2 = 108.

Now, you can insert the expressions back into the formula and find the angle. The angle between u→ and v→ becomes

cos ⁡ α = − 24 13 ⋅108 ≈−0.64, α ≈ cos ⁡ −1 (−0.64) ≈ 130.1∘.

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