Grade 10 core mathematics Measurements and Geometry – Vectors I Notes

1. (a) Identify and outline the sub-sub-strands: Vector and scalar quantities, Vector notation, Representation of vectors, Equivalent vectors, Addition of vectors, Multiplication of vectors by scalars, Column vectors, Position vectors, Magnitude of a vector, Midpoint of a vector, Translation vector.
2. (b) Explain vector and scalar quantities with illustrations.
3. (c) Use different vector notations.
4. (d) Represent vectors geometrically in different situations.
5. (e) Identify equivalent vectors in different situations.
6. (f) Add vectors in different situations (triangle and parallelogram rules).
7. (g) Multiply vectors by scalars (stretch/compress/reverse).
8. (h) Determine column vectors in different situations.
9. (i) Determine position vectors in different situations.
10. (j) Work out magnitude of a vector in different situations.
11. (k) Determine the midpoint of a vector in different situations.
12. (l) Determine translation vector as a transformation.
13. (m) Appreciate the use of vectors in real-life situations.

1. Scalars vs Vectors (Outcome b)

- Scalar: quantity with only magnitude (size). Examples: mass = 2 kg, temperature = 25 °C, time = 30 s.
- Vector: quantity with magnitude and direction. Examples: displacement 5 m east, velocity 10 m/s north.

2. Vector Notations (Outcome c)

Common ways to write the same vector:

• Bold or arrow: v or →v (in handwriting: vector AB is written as \u2192AB).
• Component form / angle brackets: <3, 4> or (3, 4).
• Column vector form: [3; 4] (a column with 3 above 4).

3. Geometric Representation (Outcome d)

A vector in the plane is shown as an arrow. The tail is starting point and the head points to the end. The arrow length shows magnitude and arrow direction shows direction.

Above: position vector OA = (3,4). Coordinates show components: x=3, y=4.

4. Equivalent Vectors (Outcome e)

Two vectors are equivalent (equal) if they have the same magnitude and direction, even if located at different places.

5. Vector Addition (Outcome f)

Two common rules:

• Triangle rule: place tail of 2nd on head of 1st; resultant is from tail of 1st to head of 2nd.
• Parallelogram rule: put tails together; resultant is diagonal of the parallelogram.

Algebraically: if p = (2,1) and q = (1,2) then p + q = (2+1, 1+2) = (3,3).

6. Scalar Multiplication (Outcome g)

Multiplying a vector by a scalar k changes its magnitude and possibly reverses direction (if k<0).

If v = (3,4) then 2v = (6,8); 0.5v = (1.5,2); -v = (-3,-4).

7. Column Vectors (Outcome h)

A column vector shows components vertically. Example: v = [3; 4] means x=3, y=4. Column vectors are convenient for matrix operations and transformations.

8. Position Vectors (Outcome i)

The position vector of point A is the vector from the origin O to A. If A has coordinates (x,y) then position vector OA = (x,y).

Example: A at (−2, 5) has OA = (−2, 5) or as column vector [−2; 5].

9. Magnitude (Length) of a Vector (Outcome j)

If v = (x, y), |v| = sqrt(x^2 + y^2) (Pythagoras).

Example: v = (3,4) → |v| = sqrt(3^2 + 4^2) = sqrt(9+16) = sqrt(25) = 5.

10. Midpoint of a Vector (Outcome k)

If A(x1,y1) and B(x2,y2), midpoint M = ((x1+x2)/2, (y1+y2)/2).

Example: A(1,2) and B(5,8) → M = ((1+5)/2, (2+8)/2) = (3,5).

11. Translation Vector as a Transformation (Outcome l)

A translation moves every point of a shape by the same vector t = (a,b). If P(x,y) is moved by t, the new point P' = (x+a, y+b).

Use in real life: moving a map marker, describing how to go from one place to another (walk 200 m east and 50 m south).

12. Real-life Uses of Vectors (Outcome m)

• Navigation and map reading (direction and distance).
• Physics: forces, velocity and acceleration (vectors add to give net force or velocity).
• Engineering and construction: describing displacements, translations of objects.
• Computer graphics: moving and scaling objects using vectors and matrices.
• Sports: describing the direction and speed of a ball or player movement.

13. Worked Examples

1. Add u = (2, −1) and v = (−1, 3).
   u + v = (2+(−1), −1+3) = (1, 2).
2. Find |w| for w = (−4, 3).
   |w| = sqrt((−4)^2 + 3^2) = sqrt(16+9) = sqrt(25) = 5.
3. Given A(2,1) and B(8,5): find AB as a column vector and midpoint M.
   AB = B − A = (8−2, 5−1) = (6, 4) = [6; 4].
   M = ((2+8)/2, (1+5)/2) = (5, 3).
4. Translate point P(3, −2) by t = (−4, 5). P' = (3−4, −2+5) = (−1, 3).

14. Suggested Learning Activities (classroom / home)

• Practical: On a graph paper (or playground grid), draw vectors for simple walks around school. Record as (x,y) and as column vectors.
• Group: Use triangle and parallelogram rules to combine several displacement vectors; check algebraic sum matches geometric resultant.
• Exercise: Find midpoint, magnitude and column vector for pairs of points around the school (e.g., classrooms, gate, field).
• Challenge: Given a small shape (triangle) on graph paper, apply translations with different vectors and check new coordinates.

15. Quick Reference (cheat-sheet)

• Notations: v, →v, (x,y), <x,y>, [x; y]
• Add: (x1,y1) + (x2,y2) = (x1+x2, y1+y2)
• Scalar mult: k(x,y) = (kx, ky)
• Magnitude: |(x,y)| = sqrt(x^2 + y^2)
• Midpoint of A and B: ((x1+x2)/2, (y1+y2)/2)
• Position vector of A(x,y): OA = (x,y)
• Translation of P by t=(a,b): P' = P + t