Writing the steps in a column
The vector a goes 4 across and 3 up. Its two components can be written stacked in a column, inside tall brackets: the step across, 4, on top, and the step up, 3, underneath. This is a column vector.
The top number is the x-component: positive to the right and negative to the left. The bottom number is the y-component: positive upward and negative downward. Written on one line inside a sentence, the same column vector is [4, 3].
Stacking the steps keeps a vector apart from a point. The pair (4, 3) on its own could name the point at x = 4, y = 3; the column says “a movement of 4 across and 3 up”, which can be made from any point.
The vector a runs 4 squares to the right and 3 squares up.
The same vector written as a column: the step across, 4, on top, and the step up, 3, underneath.
Signs give the direction
Each component can be positive or negative, so the same two numbers 4 and 3 give four different directions. [4, 3] goes right and up, [−4, 3] goes left and up, [4, −3] goes right and down, and [−4, −3] goes left and down.
All four arrows have the same length, 5, because each is the hypotenuse of a right triangle with legs 4 and 3: . Only the direction is different.
Four vectors from the origin: p = [4, 3], q = [−4, 3], r = [4, −3] and s = [−4, −3]. The signs alone set the direction.
The vector between two points
The vector from a point A to a point B is written AB with an arrow over the two letters. Its components are the end minus the start, one coordinate at a time. From A(1, 2) to B(5, 4), the step across is 5 − 1 = 4 and the step up is 4 − 2 = 2, so the vector from A to B is [4, 2].
The arrow from (1, 1) to (4, 3) is [4 − 1, 3 − 1] = [3, 2], and so is the arrow from (0, 0) to (3, 2). Arrows with the same column are the same vector, wherever they start.
The negative of a vector
The vector from B back to A is the same journey made in reverse: 4 to the left and 2 down, which is [−4, −2]. Reversing a vector changes the sign of both of its components. The reversed vector is called the negative of the vector, written with a minus sign.
For a = [4, 3], the negative is −a = [−4, −3]. It is the same arrow turned half a turn, so it has the same length, 5, and points in exactly the opposite direction.
a = [4, 3] and −a = [−4, −3]: the same length, pointing in opposite directions.
Negating a column vector changes the sign of both numbers in it.
There and back again
Make the movement a and then the movement −a, and you end where you started. In components, a + (−a) = [4 + (−4), 3 + (−3)] = [0, 0]. The vector [0, 0], with no length and no direction, is the zero vector.
b = (−3, −1)
Make the resultant lie along the y-axis
The first arrow is fixed at [3, 1], and the second is set to its negative, [−3, −1]. The second arrow leads straight back, and the resultant is (0, 0): the zero vector. Drag the head of the second arrow away from the origin, and the resultant is no longer zero.
What is not the negative
Changing only one sign does not reverse a vector. [−4, 3] points left and up, a reflection of [4, 3] in the y-axis, not its opposite.
Swapping the two components does not reverse it either. [3, 4] goes 3 across and 4 up, a different direction altogether, while [4, 3] reversed is [−4, −3].
The usual mistakes
Putting the step up on top. The top number is always the step across.
Losing a sign when an arrow points left or down. A step to the left is a negative x-component; a step down is a negative y-component.
Taking start minus end for the vector between two points. From A to B it is B minus A; A minus B is the vector from B to A, its negative.
Changing one sign, or swapping the numbers, to make the negative. The negative changes the sign of both.
Two legs of a flight
In the application below, each leg of a drone’s flight is a column vector, east on top and north underneath. Adding the two columns, top with top and bottom with bottom, gives the single vector from the base to the drone, and its length by Pythagoras is the distance straight back.
Column vectors add row by row: 1 + 3 = 4 on top and 4 + (−2) = 2 underneath.
Worked example: A Delivery Drone's Two Legs and the Flight Back to Base
Question A delivery drone leaves its base O. Its first leg is a = 25 and its second leg is b = 43, in kilometers east and north. (a) Find the displacement of the drone from its base as a column vector. (b) The drone has enough charge left for 12 km. Can it fly straight back to base, and how many kilometers of charge will it have to spare?
1.Each leg is a column vector: the top number is kilometers east and the bottom number is kilometers north. The first leg goes 2 km east and 5 km north, and the second goes 4 km east and 3 km north.
Each leg is a column vector, kilometers east on top and north underneath: a = 25 and b = 43. 2.Add the components: a + b = 25 + 43 = 2 + 45 + 3 = 68.
Add the components: a + b = 2 + 45 + 3 = 68. On the grid it is the third side of the triangle, from the start of a to the end of b. 3.(a) The drone is 6 km east and 8 km north of its base, so its displacement is 68.
(a) The drone is 6 km east and 8 km north of its base: its displacement is 68. 4.The straight flight back is the magnitude of this vector: √62 + 82 = √36 + 64 = √100 = 10 km.
The straight flight back is the magnitude: √62 + 82 = √100 = 10 km. 5.(b) 10 km is less than 12 km, so the drone can fly straight back, with 12 − 10 = 2 km of charge to spare. Check: the two legs are √29 ≈ 5.4 km and 5 km, and the straight line back, 10 km, is shorter than the 10.4 km flown, as it must be.
(b) 10 km is less than 12 km, so the drone can fly straight back with 12 − 10 = 2 km of charge to spare.
Answer: (a) 68 km; (b) yes: the flight back is 10 km, leaving 2 km to spare
Common mistakes
- Adding the lengths of the two legs, 5.4 + 5 = 10.4 km, and calling it the distance from base. The legs point in different directions, so their lengths do not add; only their components do.
- Working out √6 + 8 instead of √62 + 82. The magnitude is the hypotenuse of a right-angled triangle with sides 6 and 8, so each component is squared first.