Vectors and their components

You walk from $P = (1, 2)$ to $Q = (4, 6)$ . How far do you go to the right, and how far up?

A vector has a length and a direction, but no fixed place. We write it by its components.

From

P = (x_1, y_1)

to

Q = (x_2, y_2)

:

\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle

(end minus start). Two vectors with the same components are equal, wherever they start.

Find $\overrightarrow{PQ}$ for $P = (5, -1)$ and $Q = (2, 3)$ .

2 - 5 = -3

End minus start, horizontally.

3 - (-1) = 4

End minus start, vertically.

\overrightarrow{PQ} = \langle -3, 4 \rangle

Write the components.

Your turn: $P = (2, 7)$ and $Q = (6, 1)$ .

Horizontal:

4

6 − 2.

Vertical:

-6

, so

\overrightarrow{PQ} = \langle 4, -6 \rangle

1 − 7.

The vector $\langle 2, 1 \rangle$ starts at the point $(1, 1)$ . Tap the point where it ends.
The vector $\langle 2, 3 \rangle$ starts at the point $(1, -3)$ . Tap the point where it ends.
The vector $\langle 3, 1 \rangle$ starts at the point $(-2, -1)$ . Tap the point where it ends.

$P = (3, 3)$ and $Q = (8, 8)$ . Which vector is $\overrightarrow{PQ}$ ?
1. $\langle 5, 5 \rangle$
2. $\langle -5, -5 \rangle$
3. $\langle 11, 11 \rangle$
$P = (2, 2)$ and $Q = (8, 6)$ . Which vector is $\overrightarrow{PQ}$ ?
1. $\langle 6, 4 \rangle$
2. $\langle -6, -4 \rangle$
3. $\langle 10, 8 \rangle$

Three questions to check you have got it.

Vectors and components

$\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle$ : end minus start.
Same components, same vector, wherever it starts.

Watch out: start minus end gives the opposite vector.

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