Understanding Vectors: Magnitude, Direction & Operations

1. Vectors A vector is a quantity with both magnitude and direction. The velocity of a plane The velocity of the wind The force applied to a cue ball.

2. A vector can be represented by a directed line segment. Travelling 300 mph on a bearing 045° B This vector is named AB or u u A

3. B u A v C D AB = u = CD = v = These are known as column vectors.

4. The magnitude or size of a vector can be calculated using pythagoras. B u 3 A 5

5. v u The Addition of Vectors Two vectors can be added together to create the resultant. A canoe tries to cross a river. It travels at 4 km/h at right angles to the bank. The current travels down stream at 3km/h. What happens to the canoe? v u u + v Vectors are always added tip to tail.

6. then and Let B A C

7. Is called the zero vector written 0. so + = 0 Is the negative of v u u -v u - v

8. u If vector then and vector kv is parallel to vector v. Multiplication by a scalar vector u multiplied by a scalar k, where k>0, gives vector ku with the same direction as u but k times bigger. x k Hence if u = kv then u is parallel to v. Conversely if u is parallel to v then u =kv.

9. Position Vectors OA is the journey from the origin to the point A. It is known as the position vector written a. Is the position vector of the point b, written b. OB y = b – a where a and b are the position vectors of A and B AB A B a b o Example If P and Q have coordinates (4,8) and 2,3), respectively, find the components of PQ. x PQ = q - p

10. Collinearity Points are said to be collinear if they lie on a straight line. A, B and C are collinear if AB= kBC, where k is a scalar, then AB is parallel to BC. Also B is a common point to AB and BC.

11. Collinearity Example. Prove that the points A(2,4), B(8,6) and C(11,7) are collinear. So AB is parallel to BC and B is a common point  A, B and C are collinear.

12. The Section Formula 16 B M is the midpoint of the line AB. If A(5,8) and B(9,16), what are the coordinates of M? M 8 A M(7 , 12) 9 5 5

13. The Section Formula To calculate the point p that divides a vector in the ratio m:n you can use the section formula. B n m P A

14. Examples B (7,14) 3 1 P A (3,2)

15. a b Scalar Product For two vectors a and b the scalar product is is the angle between a & b. This is also known as the dot product. Please note the vectors must point away from the vertex.

16. a b Example Remember to use exact values where possible Rationalise the denominator

17. a b Special Case If a and b are perpendicular then a·b = 0. Therefore if a·b = 0 then a and b are perpendicular.

18. Component Form If our vectors are given in component form we can use a different method to calculate the scalar/dot product. If a = b = then a·b = a1b1 + a2b2 + a3b3