# Vectors: laws, linear dependence and uniqueness · Grade 12 · 5 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In Vectors: laws, linear dependence and uniqueness (Grade 12 · 5 units) you practise 6 skills: Resultant forces and displacements · The laws of the operations · Linear dependence · Vectors of a figure as combinations · Ratios by the uniqueness of a representation · Proofs with vectors.

Force, velocity and displacement are vectors: each has a size and a direction. When several forces act at one point of a body, the resultant force is the sum of their vectors, the diagonal of the parallelogram they make. For two forces at right angles, Pythagoras’ theorem gives the size of the resultant. At any other angle, the square of the size of the sum equals the sum of the squares of the sizes plus twice the product of the sizes and the cosine of the angle between the forces. A body is in equilibrium when all the forces on it add up to the zero vector.

The operations on vectors keep the laws of the operations on numbers. By the commutative law, the terms of a sum can change places. By the associative law, any pair in a sum of three vectors can be added first. The distributive law works both ways: a number times a sum of vectors is the sum of the multiples, and a sum of two numbers times a vector is the sum of the two multiples of the vector. The length of a sum, though, is not the sum of the lengths, unless the two vectors point the same way.

A vector is a linear combination of other vectors when it is their sum, each times a number. Two vectors are linearly dependent when they are parallel, and three vectors are linearly dependent when they lie in one plane. If two vectors are not parallel, every vector in their plane is a combination of them in exactly one way. This is the uniqueness of the representation: if two combinations of the same two vectors are equal, their coefficients are equal. It shows, for example, in what ratio two segments of a triangle cut each other: write the vector to the meeting point in two ways, and compare the coefficients.

A model proof that the diagonals of a parallelogram bisect each other: in the parallelogram ABCD, the point M is the midpoint of the diagonal AC. The vector AM is half the vector AC, that is, half the sum of the vectors AB and AD. By the triangle rule, the vector BM is the vector BA plus the vector AM, that is, half the difference of the vectors AD and AB. That difference is the vector BD, so the vector BM is half the vector BD. So M is also the midpoint of the diagonal BD, and the diagonals bisect each other.

**Tip:** Before you compare coefficients, check that the two vectors are not parallel: only then is the representation unique.

## What you learn

### Resultant forces and displacements

- Example: Two forces act at one point of a body, at right angles to each other. One force has size 30 newtons and the other 40 newtons. What is the size of the resultant force, in newtons? (Two arrows from one point, at right angles to each other)
- Answer: 50
- Solution: The forces are at right angles, so their scalar product is zero and the resultant is the hypotenuse of a right triangle. By Pythagoras’ theorem, its size is √(30² + 40²) = 50 newtons.

### The laws of the operations

- Example: Which law turns the expression (5u⃗ + v⃗) + w⃗ into 5u⃗ + (v⃗ + w⃗) ?
- Options: The distributive law over a sum of vectors · The associative law of multiplying by numbers · The distributive law over a sum of numbers · The associative law of addition
- Answer: The associative law of addition
- Solution: The step from (5u⃗ + v⃗) + w⃗ to 5u⃗ + (v⃗ + w⃗) uses the associative law of addition.

### Linear dependence

- Example: The vectors u⃗ and v⃗ are not parallel, and a⃗ = −3u⃗ + 5v⃗ and b⃗ = ku⃗ − 15v⃗ . For which value of k are the vectors a⃗ and b⃗ parallel?
- Answer: 9
- Solution: If the vectors are parallel, then b⃗ = t × a⃗ . Comparing the coefficients of v⃗ gives −15 = 5t , so t = −3 . Comparing the coefficients of u⃗ gives k = −3 × (−3) = 9 .

### Vectors of a figure as combinations

- Example: In the parallelogram ABCD , AB⃗ = u⃗ and AD⃗ = v⃗ . Which expression equals BA⃗ ? (A parallelogram with two sides from one corner drawn as arrows)
- Options: −v⃗ · u⃗ · −2u⃗ · −u⃗
- Answer: −u⃗
- Solution: In a parallelogram, BC⃗ = AD⃗ and DC⃗ = AB⃗ , so BA⃗ = −u⃗ .

### Ratios by the uniqueness of a representation

- Example: In the triangle ABC , the point D is the midpoint of BC , and the point E is the midpoint of AC . The segments AD and BE meet at O . What is the ratio AO : OD , in its simplest form? (A triangle and two segments from corners to the sides across, meeting inside it)
- Answer: 2 : 1
- Solution: Let AB⃗ = u⃗ and AC⃗ = v⃗ . Then AD⃗ = ½ × u⃗ + ½ × v⃗ and AE⃗ = ½ × v⃗ . Write AO⃗ = k × AD⃗ and BO⃗ = t × BE⃗ . From AO⃗ = AB⃗ + BO⃗ the coefficients of u⃗ and of v⃗ agree: ½ × k = 1 − t and ½ × k = ½ × t . The solution is k = 2/3 and t = 2/3 , so AO⃗ = (2/3) × AD⃗ and the ratio is AO : OD = 2 : 1 .

### Proofs with vectors

- Example: In the triangle ABC , the point M is the midpoint of AB and N is the midpoint of AC . To prove that MN is parallel to BC and half as long, the proof writes these equalities in order: (a missing step) , MN⃗ = ½ × BA⃗ + ½ × AC⃗ , MN⃗ = ½ × (BA⃗ + AC⃗) , and last MN⃗ = ½ × BC⃗ . Which step is missing? (A triangle and a segment joining the midpoints of two sides)
- Options: MN⃗ = MA⃗ − AN⃗ · MN⃗ = AM⃗ + AN⃗ · MN⃗ = MA⃗ + NA⃗ · MN⃗ = MA⃗ + AN⃗
- Answer: MN⃗ = MA⃗ + AN⃗
- Solution: The whole proof: MN⃗ = MA⃗ + AN⃗ , MN⃗ = ½ × BA⃗ + ½ × AC⃗ , MN⃗ = ½ × (BA⃗ + AC⃗) , and last MN⃗ = ½ × BC⃗ . So the segment is parallel to the third side and half as long. The missing step is MN⃗ = MA⃗ + AN⃗ .

## Not on the site yet

- The general definition of a vector space
- Equivalence classes (enrichment)

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 5 units topics](https://kixplo.com/il/en/math/grade-12-5u/)