# Vectors · Grade 12 · 4 units

> Vectors, Grade 12 · 4 units: What a vector is, Collinear vectors and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Vectors (Grade 12 · 4 units) you practise 6 skills: What a vector is · Adding and subtracting vectors · Collinear vectors · Combinations of 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

- Example: In the parallelogram ABCD , which vector equals BA⃗ ? (A parallelogram with its corners marked by letters and an arrow on one side)
- Options: DC⃗ · BC⃗ · BD⃗ · CD⃗
- Answer: CD⃗
- Solution: 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

- Example: In the triangle ABC , which vector equals CB⃗ + BA⃗ ? (A triangle with two of its sides drawn as arrows)
- Options: AC⃗ · BC⃗ · AB⃗ · CA⃗
- Answer: CA⃗
- Solution: 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

- Example: The vectors satisfy u⃗ = 3v⃗ , and neither vector is the zero vector. Are u⃗ and v⃗ collinear? (Two parallel arrows standing for the two vectors)
- Options: 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: Yes, because one is a number times the other
- Solution: 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.

### Combinations of vectors

- Example: In the triangle ABC , the point M is the midpoint of BC . Which expression equals AM⃗ ? (A triangle with a segment from a corner to the midpoint of the side across, and two sides drawn as arrows)
- Options: AB⃗ + AC⃗ · ½ × (AB⃗ − AC⃗) · ½ × (AC⃗ − AB⃗) · ½ × (AB⃗ + AC⃗)
- Answer: ½ × (AB⃗ + AC⃗)
- Solution: The point M is reached two ways: AM⃗ = AB⃗ + BM⃗ and AM⃗ = AC⃗ + CM⃗ . The vectors BM⃗ and CM⃗ are opposite, so adding the two gives 2 × AM⃗ = AB⃗ + AC⃗ , that is AM⃗ = ½ × (AB⃗ + AC⃗) .

### Vectors in coordinates

- Example: Where does the point P(0, 0, 4) lie? (Three axes in space leaving the origin)
- Options: On the x axis · On the y axis · In the xy plane · On the z axis
- Answer: On the z axis
- Solution: Two coordinates of the point are zero, x = y = 0 , so it lies on the z axis.

### Working in coordinates

- Example: The vectors u⃗ = (−7, 9, −7) and v⃗ = (0, 7, 4) are given. What is u⃗ + v⃗ ? (Three axes in space leaving the origin)
- Answer: (−7, 16, −3)
- Solution: Add component by component: u⃗ + v⃗ = (−7, 16, −3) .

## Not on the site yet

- Proofs with vectors

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 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 4 units topics](https://kixplo.com/il/en/math/grade-12-4u/)