Vectors
Senior 2 · Mathematics
Vectors
Lesson 3 of 4
22 min
Uwase walks from home to the market, then from the market to school. However winding the two legs are, her overall change of position is one vector: home to school. A boat crossing Lake Kivu is pushed by its engine and by the wind at the same time, and it moves along the combination of the two. Adding vectors answers the question "where do I end up?"
To add a and b, draw b starting from the head of a. The sum a + b (also called the resultant) goes straight from the tail of a to the head of b.
Figure 3. The triangle rule: a = (3, 1) followed by b = (1, 3) gives a + b = (4, 4). Diagram: Ganzaa.
With points this reads AB + BC = AC (the Chasles relation): going from A to B and then from B to C is the same as going from A to C.
If you draw a and b from the same point instead, the sum is the diagonal of the parallelogram they make: the parallelogram rule. It gives the same answer.
Add the top numbers together and the bottom numbers together. In Figure 3: 31 + 13 = 44.
To subtract, add the opposite: a − b = a + (−b). With components, subtract top from top and bottom from bottom.
If k is a number (a scalar), ka multiplies both components by k: kxy = kxky.
Two vectors are parallel when one is a scalar multiple of the other. 6−9 = 32−3, so these are parallel. 46 is not parallel to 2−3: 4 = 2 × 2 but 6 ≠ 2 × ( − 3).
a = 3−1 and b = −24. Find a + b, a − b and 2a − 3b.
Mugisha walks 23 (in hundreds of metres) from home to a shop, then 4−1 from the shop to school. His overall displacement is 23 + 4−1 = 62: 600 m east and 200 m north of home.
With a and b from Worked example 1, find the vector c such that a + c = b.
Rearrange: c = b − a = −2 − 34 − ( − 1) = −55.
Tip
Cancel the middle letter. In a chain of vectors the inner letters must match, and they drop out: PQ + QR + RS = PS. If the letters do not link up, turn a vector round first: −CB = BC.
Brackets are your friend with negatives: 3 − ( − 2) = 5, not 1.
Use p = 4−2 and q = −15.
Find p + q.
Solution: 4 − 1−2 + 5 = 33.
Find p − q.
Solution: 4 − ( − 1)−2 − 5 = 5−7.
Find 3p + 2q.
Solution: 12−6 + −210 = 104.
Simplify (a) AB + BC + CD; (b) AB − CB.
Solution: (a) AD. (b) −CB = BC, so AB + BC = AC.
Find k and m if k2−1 = m3.
Solution: Bottom components: −k = 3, so k = −3. Top components: m = 2k = −6.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
