➡️Vectors

Loremaster Cos keeps the charts of the Hall of Laws in arrows rather than in dots. A dot answers one question — where — and a number answers one question — how much. An arrow answers two at once: how much, and which way.

That is what a vector is: a quantity with a magnitude (a size, never negative) and a direction. A wind of 30 km/h from the west, a pull of 12 newtons down the slope, a walk of 5 km north-east: each of them needs both numbers, and neither number alone is the whole story.

Here is the part that surprises everyone. A vector does not record where it was drawn. Slide the arrow anywhere you like on the page and, as long as you do not turn it or stretch it, it is the *same vector*. Two arrows are equal when they have the same length and point the same way — nothing else is checked.

So how do you write one down? By recording the journey, not the ends of it: how far across, then how far up. That pair is the component form of the vector.

v = (4, 3) means: travel 4 to the right, then 3 up

Both components may be negative, and a negative one simply means the travel runs the other way — left instead of right, or down instead of up.

Reading a vector off two points. If an arrow runs from an initial point P to a terminal point Q, its components are

PQ = (x of Q - x of P, y of Q - y of P)

Terminal minus initial, in that order and in both slots. For P(2, 7) and Q(-1, 3):

PQ = (-1 - 2, 3 - 7) = (-3, -4)

Run the subtraction the other way and you get (3, 4), which is the arrow QP — the same road travelled backwards. The two are not the same vector, so the order is worth saying out loud every time.

The rule also runs in reverse. Since PQ = Q - P, rearranging gives Q = P + PQ and P = Q - PQ, which is how you find a missing end of an arrow when you are handed the other end and the components.

Every vector carries two measurements, and each has its own formula.

Magnitude. Draw the arrow from the origin and drop a right triangle under it: one leg runs across, one leg runs up, and the arrow itself is the hypotenuse. Pythagoras does the rest.

|v| = sqrt(x^2 + y²)

For v = (-3, -4): |v| = sqrt(9 + 16) = sqrt(25) = 5. Notice that the minus signs disappeared. Squaring makes both components positive, so a magnitude is never negative — an arrow pointing down and to the left is exactly as long as its mirror image pointing up and to the right.

Sometimes the answer is whole, because the components happen to be two legs of a Pythagorean triple (3-4-5, 5-12-13, 8-15-17, 7-24-25). Usually it is not, and then it is left as a square root or rounded, as the question asks.

Direction angle. The direction angle t is measured anticlockwise from the positive x-axis, and it lives on [0°, 360°). Its tangent comes straight from the components:

tan t = y / x

but here is the trap that this whole topic turns on. A calculator's arctan only ever answers between -90° and 90°, because tangent repeats every 180°. It genuinely cannot tell the arrow (3, 4) from the arrow (-3, -4) — the ratio y/x is the same for both. So arctan gives you the reference angle, and you supply the quadrant:

• ◆pointing right and up (quadrant I): the answer is the arctan value itself
• ◆pointing left, up or down (quadrants II and III): add 180°
• ◆pointing right and down (quadrant IV): add 360° to the negative arctan value

For v = (-6, 8): arctan(8 / -6) hands back -53.13°. The arrow points up and to the left, into quadrant II, so the direction angle is -53.13 + 180 = 126.87°, about 127°.

Sketch the arrow before you touch a calculator. Ten seconds of drawing settles the quadrant, and the quadrant is the half of the answer no formula will give you.

An arrow lying along an axis needs no arctangent at all: (5, 0) is 0°, (0, 5) is 90°, (-5, 0) is 180° and (0, -5) is 270°. Read those straight off the sketch.

Adding two vectors means doing one journey and then the other. Draw u, then start v at the tip of u; the arrow from the very beginning to the very end is u + v. That is the tip-to-tail picture, and it is why a vector sum is called a resultant.

In components it is far easier than the picture suggests: the across parts join the across parts, the up parts join the up parts, and nothing ever crosses between the two slots.

(5, 2) + (-1, 4) = (5 + (-1), 2 + 4) = (4, 6)

Subtracting works slot by slot in exactly the same way, and order matters: u - v and v - u are opposite arrows of the same length.

(5, 2) - (-1, 4) = (6, -2)

One warning that catches everyone at least once. Magnitudes do not add. |u + v| is almost never |u| + |v| — the only time it is, is when u and v point the same way. Two pulls of 3 and 4 at right angles give a resultant of 5, not 7. Add the *components*, then measure once at the end.

Because addition and subtraction behave, so do vector equations. Solve u + w = v exactly as you would solve 3 + w = 7:

u + w = v => w = v - u

Scalar multiplication stretches an arrow without turning it. Multiplying by a number k multiplies every component:

k(a, b) = (ka, kb)

The effect on the picture is easy to state. |k| decides the new length: |ku| = |k| |u|. The *sign* of k decides the direction: a positive k keeps the arrow pointing the same way, and a negative k turns it round to point the opposite way. So 3u is three times as long the same way, and -2u is twice as long the other way.

Two vectors are parallel exactly when one is a scalar multiple of the other. To test a pair, divide matching components: (6, -9) and (-4, 6) give -4/6 and 6/-9, which are both -2/3, so they are parallel. If the two divisions disagree, the arrows lie on different lines however close their numbers look.

A linear combination such as 3u - 2v is just these two rules used in order: scale each vector first, then combine. And the minus sign belongs to the whole of 2v, so both of its components are taken away.

Component form and the magnitude-and-direction description are two ways of saying the same thing, and you should be able to walk between them in either direction.

Going one way, you already have the tools: magnitude by Pythagoras, direction by arctan plus the quadrant.

Going the other way, put the arrow on the origin and read the right triangle again. The across leg is the adjacent side and the up leg is the opposite side, so

v = (|v| cos t, |v| sin t)

Cosine builds the across component, sine builds the up one — every time, in every quadrant. There is no separate rule for quadrant III; the cosine and sine simply come out negative there and carry their signs into the components.

A worked instance. A pull of magnitude 10 acts at a direction angle of 150°.

x = 10 cos 150 = 10 * (-sqrt(3)/2) = -5 sqrt(3) y = 10 sin 150 = 10 * (1/2) = 5

So the pull is (-5 sqrt(3), 5): about (-8.7, 5). The negative x and positive y agree with the picture, because 150° points up and to the left.

At the special angles 30°, 45°, 60° and their partners round the circle, one component usually comes out as a whole number times a square root, and that exact form is worth keeping. Round only at the very end, and never round a ratio on the way through — rounding cos 30 to 0.87 and then multiplying costs you accuracy you did not have to lose.

There is a shortcut worth spotting. If a question hands you cos t and sin t directly (say cos t = 3/5 and sin t = -4/5), you do not need the angle at all: multiply the magnitude by each ratio and you have the components.

A unit vector is a vector of magnitude exactly 1. It carries a direction and nothing else — all the length has been divided out.

To find the unit vector pointing the same way as v, divide v by its own magnitude:

unit vector = v / |v| = (x / |v|, y / |v|)

For v = (3, 4): |v| = 5, so the unit vector is (3/5, 4/5). Check it: (3/5)² + (4/5)² = 9/25 + 16/25 = 1. It works out every time, because dividing both components by |v| divides the whole length by |v|.

Both components are divided, and they are divided by the magnitude — not by one of the components, and not by |v| squared. A unit vector always has components between -1 and 1, which is a quick sanity check on your answer.

This gives a neat recipe for building a vector to order. To make a vector of magnitude 15 in the direction of (-3, 4): the magnitude of (-3, 4) is 5, so the unit vector is (-3/5, 4/5), and 15 times that is (-9, 12). Shrink to 1, then stretch to the length you want.

i and j. Two unit vectors are used so often that they have names:

i = (1, 0) one step to the right j = (0, 1) one step up

Every vector is a combination of those two, which gives the other standard notation:

(3, -2) = 3i - 2j 5i + j = (5, 1)

It is the same vector wearing different clothes. The number in front of i is the across component and the number in front of j is the up component, in that order, with the sign belonging to its own term. A term with no number in front carries a 1.

Everything in this lesson works identically in i, j notation — you add the i parts, add the j parts, and a scalar multiplies both — so read the components off first and carry on as usual.