Vectors: laws, linear dependence and uniqueness · Years 11 and 13
In Vectors: laws, linear dependence and uniqueness (Years 11 and 13) you practise 3 skills: The laws of the operations · Vectors of a figure as combinations · Ratios by the uniqueness of a representation. 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
The laws of the operations
For example: Which law turns the expression (5u⃗ + v⃗) + w⃗ into 5u⃗ + (v⃗ + w⃗) ?
- 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 and solution
The answer: The associative law of addition
The step from (5u⃗ + v⃗) + w⃗ to 5u⃗ + (v⃗ + w⃗) uses the associative law of addition.
Vectors of a figure as combinations
For example: In the parallelogram ABCD , AB⃗ = u⃗ and AD⃗ = v⃗ . Which expression equals BA⃗ ?
- −v⃗
- u⃗
- −2u⃗
- −u⃗
Answer and solution
The answer: −u⃗
In a parallelogram, BC⃗ = AD⃗ and DC⃗ = AB⃗ , so BA⃗ = −u⃗ .
Ratios by the uniqueness of a representation
For 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?
Answer and solution
The answer: 2 : 1
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 .
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4) and selected AS and A level mathematics practice in England (the GCE subject content).