Vector Arithmetic

Main idea

To add vectors, put them tip to tail; to scale one, stretch it by a number. Both turn out to be as easy as doing the arithmetic one component at a time.

\vec a + \vec b = \langle a_1 + b_1,\; a_2 + b_2,\; a_3 + b_3\rangle, \qquad {\htmlData{sym=0}{c}}\,\vec a = \langle {\htmlData{sym=0}{c}} a_1,\; {\htmlData{sym=0}{c}} a_2,\; {\htmlData{sym=0}{c}} a_3\rangle

What the symbols mean

$c$
Scalar
A plain number. It stretches the vector by |c| and flips it if c < 0.

Explanation

Vector arithmetic is simply the rules for combining trips: adding two vectors means taking one trip and then the other, and scaling a vector means making the same trip longer, shorter or backward. It's what lets you combine a boat's speed with a river's current, or add up every force on a bridge. The formula $\vec a + \vec b = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3\rangle$ looks busy, but it's just "add the east parts, add the north parts, add the up parts." And $c\,\vec a$ (the same trip, $c$ times as long) just multiplies each part by $c$ . It's ordinary arithmetic done three times.

Adding. Think of vectors as moves. If you walk along $\vec a$ and then along $\vec b$ , where do you end up? Put the tail of $\vec b$ at the tip of $\vec a$ ; the sum runs from the start of $\vec a$ to the end of $\vec b$ . Do the moves in the other order and you trace the other two sides of a parallelogram, landing in the same place, so the sum is that parallelogram's diagonal. It makes sense that this works component by component: the east parts of the two moves add up, the north parts add up, and so on.

Subtracting is adding the opposite, $\vec a - \vec b = \vec a + (-\vec b)$ . Picture it as the vector from the tip of $\vec b$ to the tip of $\vec a$ , the same "end minus start" idea from the last lesson.

Scaling. Multiplying by $c$ multiplies every component by $c$ , so the length gets multiplied too: $\|c\,\vec a\| = |c|\,\|\vec a\|$ . The direction stays the same, or flips if $c$ is negative. That gives a clean test: two vectors are parallel when one is a scalar multiple of the other.

Linear combinations like $2\vec a - 3\vec b$ mix these two operations, and they're the basic building blocks of linear algebra. For instance, every vector in the plane is some combination of $\vec i$ and $\vec j$ .

See it

A boat doing 3 m/s through the water, in a 1.2 m/s current. Its real velocity over the ground is the two added tip to tail. As the heading swings around, notice how much the current matters: heading straight into it leaves just 1.8 m/s, and going with it gives 4.2 m/s.

Worked examples

Example 1

With $\vec a = \langle 4, 0, 3\rangle$ and $\vec b = \langle -1, 2, 5\rangle$ , find $2\vec a - 3\vec b$ .

1
Scale first: $2\vec a = \langle 8, 0, 6\rangle$ and $3\vec b = \langle -3, 6, 15\rangle$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find $\langle 1, 2\rangle + \langle 3, -1\rangle$ .

Problem 2+5 XP

Find $3\langle 2, -1, 0\rangle - \langle 1, 1, 1\rangle$ .

Problem 3+5 XP

Find $\|-2\langle 3, 4\rangle\|$ .

Problem 4+5 XP

Are $\langle 2, -4, 1\rangle$ and $\langle -6, 12, -3\rangle$ parallel?

Problem 5+5 XP

Find $a$ and $b$ with $a\langle 1, 0\rangle + b\langle 1, 1\rangle = \langle 5, 3\rangle$ . Answer as $a, b$ .

Problem 6+5 XP

A boat moves at $3$ m/s through the water, in a river whose current flows east at $1.2$ m/s. The pilot wants to go straight north, across the river. How fast does the boat actually make progress north?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A $50$ N sign hangs from two cables attached to the ceiling: the left one makes a $30°$ angle with the ceiling, the right one $45°$ . Find the tension in the right-hand cable.

Where it's used

Navigation adds vectors constantly: a plane's ground velocity is its air velocity plus the wind's. Net force in physics is the vector sum of every force.

Vectors: The Basics The Dot Product