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Geometry & measures · Vectors

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Column vectors

A vector is a translation — so far across, so far up — not a place. How to add, subtract and scale columns, why a scalar multiple is parallel, how Pythagoras gives the magnitude, and why the journey from A to B is destination minus start.

Geometry & measures · Vectors

Column vectors

A vector is a translation — so far across, so far up — not a place. How to add, subtract and scale columns, why a scalar multiple is parallel, how Pythagoras gives the magnitude, and why the journey from A to B is destination minus start.

Why it works

A vector is not a place. It is a move. The column vector

(3−2)\binom{3}{-2}

is one instruction: go 33 across, then 22 down. The top number is the movement in the xx direction (positive means right, negative means left) and the bottom number is the movement in the yy direction (positive means up, negative means down). That is the entire definition, and everything else in this topic falls out of it.

A vector has no starting point. This is the part students find strangest, so make it concrete. Start at the origin and obey (3−2)\binom{3}{-2}: you land on (3,−2)(3, -2). Start at (−4,1)(-4, 1) and obey the same instruction: you land on (−1,−1)(-1, -1). Start at (0,4)(0, 4): you land on (3,2)(3, 2). Three different journeys ending in three different places — but one single vector, because "3 across, 2 down" is the same instruction every time.-6-5-4-3-2-1123456-3-2-112345aaaxySo (3−2)\binom{3}{-2} and the coordinate (3,−2)(3, -2) are different kinds of object, even though they carry the same two numbers. The coordinate answers where? and is nailed to one spot on the grid. The column vector answers how do I get from here to there? and can be drawn anywhere on the grid. Two arrows are the same vector exactly when they have the same length and point the same way — where they happen to sit on the page is irrelevant.

Adding is doing one journey and then the other. Obey (51)\binom{5}{1}, and then, from wherever you have ended up, obey (−14)\binom{-1}{4}. How far across have you moved altogether? 5+(−1)=45 + (-1) = 4. How far up? 1+4=51 + 4 = 5. So the single instruction with the same overall effect — the resultant — is (45)\binom{4}{5}:

(51)+(−14)=(5+(−1)1+4)=(45).\binom{5}{1} + \binom{-1}{4} = \binom{5 + (-1)}{1 + 4} = \binom{4}{5}.

Adding row by row works because sideways movement and upward movement do not interfere with each other: however much you shuffle left and right, your height is untouched, and vice versa. Drawn nose to tail, the two journeys and the resultant close up into a triangle.aba + bSubtracting is the same rule with a minus sign, and it means undo: a−b=a+(−b)\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b}), so do a\mathbf{a} and then walk b\mathbf{b} backwards. The bottom row is a subtraction too, which is where signs go wrong:

(2−3)−(5−8)=(2−5−3−(−8))=(−35).\binom{2}{-3} - \binom{5}{-8} = \binom{2 - 5}{-3 - (-8)} = \binom{-3}{5}.

Take away a negative and the total goes up — −3−(−8)=−3+8=5-3 - (-8) = -3 + 8 = 5.

Multiplying by a number stretches the journey. 2a2\mathbf{a} means "do a\mathbf{a} twice", so both rows double: if a=(3−2)\mathbf{a} = \binom{3}{-2} then 2a=(6−4)2\mathbf{a} = \binom{6}{-4}, not (6−2)\binom{6}{-2}. Doubling only the top would change the shape of the journey — it would be a different direction, not the same journey done twice.

Why is 2a2\mathbf{a} parallel to a\mathbf{a}? Because the direction of an arrow is fixed by the ratio of across to up, and doubling both numbers leaves that ratio alone: 6:−46 : -4 is the same slope as 3:−23 : -2. And it is exactly twice as long, because 62+(−4)2=4(32+(−2)2)=232+(−2)2\sqrt{6^2 + (-4)^2} = \sqrt{4(3^2 + (-2)^2)} = 2\sqrt{3^2 + (-2)^2} — pulling a factor of 22 out of the square root. Meanwhile −a=(−32)-\mathbf{a} = \binom{-3}{2} is that same journey walked backwards: the same length, the exact opposite direction. In general kak\mathbf{a} is parallel to a\mathbf{a}, is ∣k∣|k| times as long, and points the same way when k>0k > 0 and the opposite way when k<0k < 0. Read backwards, that is the GCSE test for parallel: if one vector is a scalar multiple of another, the two are parallel.

Length comes from Pythagoras. The two numbers in the column are the two short sides of a right-angled triangle, and the vector itself is the hypotenuse. So the magnitude of a=(xy)\mathbf{a} = \binom{x}{y} is

∣a∣=x2+y2.|\mathbf{a}| = \sqrt{x^2 + y^2}.

For example ∣(4−3)∣=42+(−3)2=16+9=25=5\left|\binom{4}{-3}\right| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5.a4 across3 downSquaring destroys the sign, and that is exactly right: (4−3)\binom{4}{-3} and (−43)\binom{-4}{3} point opposite ways but are both 55 long. So never write 42−32\sqrt{4^2 - 3^2} just because a component is negative — (−3)2=+9(-3)^2 = +9.

The translation from one point to another. To get from A(1,4)A(1, 4) to B(6,2)B(6, 2): across, 6−1=56 - 1 = 5; up, 2−4=−22 - 4 = -2. So AB→=(5−2)\overrightarrow{AB} = \binom{5}{-2} — destination minus start. Do the subtraction the other way round and you get (−52)\binom{-5}{2}, which is BA→\overrightarrow{BA}, the journey home. There is a check that catches this every single time: start at AA and obey your own answer. (1,4)(1, 4), then 55 across and 22 down, lands on (6,2)(6, 2), which is BB. ✓ If you land anywhere else, you subtracted the wrong way round.