Vector Voyage

Column vectors

A vector is a movement: a direction together with a distance. It is written as a column of two numbers, and which number is which never changes: top = across, bottom = up. So (3, 2) means 3 right then 2 up. A negative reverses that component: −3 on top means 3 left, and −2 on the bottom means 2 down. One thing to hold on to throughout: a vector says nothing about where you started. It is the journey, not the destination, which is why the same vector can be drawn anywhere on a grid.

Which vector gets you there?

You are standing at the point (5, 2) and you want to reach the point (1, 4). Which column vector describes that journey?

• (−4, 2)
• (4, −2)
• (6, 6)
• (2, −4)

Vectors and scalars

The distinction runs underneath everything in this topic, and it explains why vectors have their own arithmetic:

Three legs of a journey

A walker follows three vectors in turn: (2, 1), then (3, 4), then (−1, 2). What is the BOTTOM number of the single vector describing the whole journey?

Multiplying and subtracting

Two operations, and one common misconception worth clearing up:

What does −2 do?

A vector v is (3, 1). What is −2v, and how does it compare with v?

• (−6, −2): twice as long as v and pointing in exactly the opposite direction
• (−6, −2): twice as long as v and pointing the same way, since multiplying only changes the length
• (−6, −2): half the length of v and in the opposite direction
• (−6, 1): only the top component is affected by the multiplication

Subtract them

Work out (5, 3) − (2, 7). What is the BOTTOM number of the answer?

Four things that go wrong

Never mix the rows. The top number only ever meets other top numbers. If a top has met a bottom in your working, something has gone wrong, whatever the arithmetic said. Subtraction is not reversible. a − b and b − a are different answers. Work out which order the question wants before you start, and for a journey it is always finish minus start. A magnitude is never negative. Both components are squared, and squaring removes the sign, so |(−6, 8)| and |(6, 8)| are identical. A negative answer to a magnitude means an arithmetic slip, not a vector pointing backwards. Parallel does not mean equal. (8, 6) is parallel to (4, 3), and it is twice as long. To show two vectors are parallel, show one is a scalar multiple of the other, and say what the scalar is.

Which is parallel?

Which of these vectors is parallel to (2, 3)?

• (6, 9)
• (3, 2)
• (−2, 3)
• (4, 5)

How long is a vector?

The magnitude of a vector is its straight-line length, written with vertical bars: |v|. Because the across and up parts are at right angles to each other, a vector and its two components form a right-angled triangle, and the magnitude is the hypotenuse. |(5, 12)| = √(5² + 12²) = √(25 + 144) = √169 = 13. Square each component, add them, take the square root. That is Pythagoras with no changes at all, and it is why 3-4-5 and 5-12-13 turn up so often in vector questions. And signs make no difference. |(−5, 12)| is also 13, because (−5)² is 25 exactly as 5² is. A magnitude is a length, so it is always positive whichever way the vector points.

Find the magnitude

Find the magnitude of the vector (−6, 8).

Plot where you land

You start at the point (1, 1), already plotted, and apply the translation vector with 3 on top and 2 on the bottom. Plot the point you arrive at.

What is each vector doing?

The voyage rules

In a column vector the top number is the movement _____ and the bottom number is the movement up, and the two are never mixed. To find the vector from one point to another, take the finish and subtract the _____. Multiplying by a positive scalar changes only the length, while multiplying by a negative scalar also reverses the _____. Two vectors are parallel when one is a scalar multiple of the other. And the magnitude is found with Pythagoras, so it is always _____, whichever way the vector points.