Vectors
Senior 2 · Mathematics
Vectors
Lesson 4 of 4
20 min
A delivery drone takes off from a distribution centre and flies 6 km east and then 8 km north to reach a health centre. How far is the health centre in a straight line, and how long would a direct flight take? The answer is the magnitude of the displacement vector, and you already have the tool for it: Pythagoras' theorem.
The components of a = xy are the two shorter sides of a right-angled triangle, and a is the hypotenuse. So
|a| = √(x² + y²)
Worked example 1
The drone's displacement is 68 km. Its magnitude is √(6² + 8²) = √(36 + 64) = √100 = 10 km.
More examples: |−512| = √(25 + 144) = √169 = 13, and |2−3| = √(4 + 9) = √13 ≈ 3.61.
The distance AB is the magnitude of AB. For A(x1, y1) and B(x2, y2):
AB = √((x2 − x1)² + (y2 − y1)²)
Worked example 2
A(−2, 1) and B(4, 9). AB = 68, so AB = √(36 + 64) = 10 units.
The midpoint M of AB is halfway along AB, so AM = 12 AB. This gives the midpoint formula: average the coordinates.
M = ((x1 + x2) ÷ 2, (y1 + y2) ÷ 2)
Worked example 3
For A(−2, 1) and B(4, 9): M = (( − 2 + 4) ÷ 2, (1 + 9) ÷ 2) = (1, 5). Check: AM = 34, exactly half of AB = 68.
Worked example 4
(finding an end point). M(3, −1) is the midpoint of PQ and P is (7, 2). Find Q.
Since M is halfway, Q = (2 × 3 − 7, 2 × ( − 1) − 2) = ( − 1, −4).
Divide a vector by its own magnitude and you get a vector of length 1 pointing the same way. For a = 34, |a| = 5, so the unit vector is 0.60.8. Check: √(0.36 + 0.64) = √1 = 1.
On a map of Lake Kivu drawn with 1 unit = 1 km, a boat leaves (1, 2) for an island at (9, 8). The trip is 86, so the straight-line distance is √(64 + 36) = 10 km. At 20 km/h the crossing takes 10 ÷ 20 = 0.5 h, which is 30 minutes.
Tip
Square, add, root. Square each component, add, then take the square root. Squares are never negative: (−5)² = 25, not −25.
Do not add the components themselves: 6 + 8 = 14 is the distance along the two legs, not the straight-line distance of 10.
Find |912|.
Solution: √(81 + 144) = √225 = 15.
Find |−724|.
Solution: √(49 + 576) = √625 = 25.
Find the distance between P(1, −2) and Q(6, 10).
Solution: PQ = 512, so PQ = √(25 + 144) = √169 = 13.
Find the midpoint of C(−3, 8) and D(5, −2).
Solution: (( − 3 + 5) ÷ 2, (8 − 2) ÷ 2) = (1, 3).
M(2, 5) is the midpoint of AB and A is (−1, 1). Find B and the length AB.
Solution: B = (2 × 2 − ( − 1), 2 × 5 − 1) = (5, 9). AB = 68, so AB = 10.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
