Vectors · Years 11 and 13
In Vectors (Years 11 and 13) you practise 5 skills: What a vector is · Adding and subtracting vectors · Collinear vectors · Vectors in coordinates · Working in coordinates. A vector has a magnitude and a direction, and is drawn as an arrow. Two vectors are equal when they have the same magnitude and the same direction, even if they start at different points. In a parallelogram, for example, the sides AB and DC stand for equal vectors. The opposite vector has the same magnitude and the opposite direction, and a vector plus its opposite is the zero vector.
By the triangle rule, the second vector starts at the end of the first, and the sum is the arrow from the start of the first to the end of the second. By the parallelogram rule, both vectors start at the same point, and the sum is the diagonal of the parallelogram they make. Subtracting is adding the opposite. Addition is commutative and associative, so three vectors give the same sum in any order, and multiplying by a number distributes over addition: twice the sum of two vectors is the sum of their doubles. Two vectors that are not the zero vector are collinear when one is a number times the other. They may have different magnitudes and point opposite ways. Without proof: if two vectors in a plane are not collinear, every vector in the plane is a combination of them in exactly one way. In the same way, every vector in space is exactly one combination of three vectors that are not in one plane. Space has three axes through the origin, and every point has three coordinates. The components of the vector from one point to another are the second point’s coordinates minus the first’s. The unit vectors have magnitude one along the three axes, and every vector is a combination of them. In coordinates, vectors are added component by component, and a vector is multiplied by a number by multiplying each component.
Tip: Check each answer on a drawing: sketch the arrows, and make sure the sum starts and ends in the right places.
What you learn
What a vector is
For example: In the parallelogram ABCD , which vector equals BA⃗ ?
- DC⃗
- BC⃗
- BD⃗
- CD⃗
Answer and solution
The answer: CD⃗
The side CD is parallel to BA and has the same length, and the vector CD⃗ points the same way. So BA⃗ = CD⃗ .
Adding and subtracting vectors
For example: In the triangle ABC , which vector equals CB⃗ + BA⃗ ?
- AC⃗
- BC⃗
- AB⃗
- CA⃗
Answer and solution
The answer: CA⃗
The second vector starts at B , where the first one ends. By the triangle rule, the sum runs from the start of the first to the end of the second: CB⃗ + BA⃗ = CA⃗ .
Collinear vectors
For example: The vectors satisfy u⃗ = 3v⃗ , and neither vector is the zero vector. Are u⃗ and v⃗ collinear?
- No, because their magnitudes are different
- No, because collinear vectors must point the same way
- No, because they do not start at the same point
- Yes, because one is a number times the other
Answer and solution
The answer: Yes, because one is a number times the other
Here u⃗ is a number times v⃗ , so they are collinear, pointing the same way. The magnitudes may differ, and the vectors need not start at the same point.
Vectors in coordinates
For example: Where does the point P(0, 0, 4) lie?
- On the x axis
- On the y axis
- In the xy plane
- On the z axis
Answer and solution
The answer: On the z axis
Two coordinates of the point are zero, x = y = 0 , so it lies on the z axis.
Working in coordinates
For example: The vectors u⃗ = (−7, 9, −7) and v⃗ = (0, 7, 4) are given. What is u⃗ + v⃗ ?
Answer and solution
The answer: (−7, 16, −3)
Add component by component: u⃗ + v⃗ = (−7, 16, −3) .
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).