Resolving vectors and unit vectors

A pull that does two jobs

You pull a suitcase along an airport floor by its handle. The handle slopes up towards your hand. Part of your pull moves the suitcase forward. Another part tries to lift it. How much of each?

To answer, we split the pull into two parts: one along the floor and one straight up. This is called resolving the vector, and the parts are its components. It is the most useful idea in this chapter, because it turns every vector problem into two simple problems along two straight lines.

Finding the two parts

Put the tail of $\vec{A}$ at the origin. Let it make the angle $\theta$ with the $+x$ axis. Drop a line from its head down to the $x$ axis. You get a right triangle, and its two short sides are the components.

Why cos for $x$? The $x$ part is the side next to the angle $\theta$, and cos goes with the side next to the angle. This is only true when $\theta$ is measured from the $x$ axis. If a question gives the angle from the vertical, the jobs swap: the $y$ part takes the cos.

Going back from the parts to the whole vector:

Signs and quarters

Components carry signs. A part that points along $-x$ is negative. Move the slider round the full circle and watch.

Here is the trap. $\tan\theta$ is the same for $(3, 4)$ and $(-3, -4)$, because $\dfrac{4}{3} = \dfrac{-4}{-3}$. But one arrow points up and to the right, and the other points down and to the left. A calculator's $\tan^{-1}$ gives $53.1^\circ$ for both.

So always do two steps:

1. Find the angle with the $x$ axis from $\tan^{-1}\left|\dfrac{A_y}{A_x}\right|$.
2. Use the signs of $A_x$ and $A_y$ to place the arrow in its quarter.

Example. $\vec{A} = -3\hat{i} + 4\hat{j}$. The size is $\sqrt{9 + 16} = 5$. The angle with the $x$ axis is $\tan^{-1}\dfrac{4\ \text{km}}{3\ \text{km}} = 53.1^\circ$. The $x$ part is negative and the $y$ part is positive, so the arrow points up and to the left. Its angle from $+x$ is $180^\circ - 53.1^\circ = 126.9^\circ$.

In Python, math.atan2(Ay, Ax) looks at both signs and does step 2 for you.

Unit vectors along the axes

We give the two axes their own unit vectors: $\hat{i}$ (read "i hat") along $+x$, and $\hat{j}$ along $+y$. Each has size $1$.

Now any vector in the plane can be written with its two parts:

For example, $\vec{r} = 4\hat{i} + 3\hat{j}$ means "go $4$ along $x$, then $3$ along $y$". Its size is $\sqrt{16 + 9} = 5$. In three dimensions we add $\hat{k}$ along $z$.

Adding vectors the easy way

With components, adding needs no drawing at all. If $\vec{A} = A_x\hat{i} + A_y\hat{j}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j}$, then

Add the $x$ parts. Add the $y$ parts. Keep them apart. Subtraction and multiplying by a number work the same way, one part at a time.

Worked example. A hiker walks $6\ \text{km}$ east, then $4\ \text{km}$ north, then $5\ \text{km}$ towards the south-west ($225^\circ$ from east). Where does she end up?

1. Split each walk. East: $(6, 0)$. North: $(0, 4)$. South-west: $(5\cos 225^\circ, 5\sin 225^\circ) = (-3.54, -3.54)$.
2. Add the parts. $R_x = 6\ \text{km} + 0\ \text{km} - 3.54\ \text{km} = 2.46\ \text{km}$ and $R_y = 0\ \text{km} + 4\ \text{km} - 3.54\ \text{km} = 0.46\ \text{km}$.
3. Put them back together. $R = \sqrt{(2.46\ \text{km})^2 + (0.46\ \text{km})^2} = 2.51\ \text{km}$, at $\tan^{-1}\dfrac{0.46\ \text{km}}{2.46\ \text{km}} = 10.7^\circ$ north of east.
4. Check: both totals are positive, so she ends up to the north-east of the start. That fits the small angle.

Back to the suitcase. A pull of $80\ \text{N}$ along a handle at $40^\circ$ to the floor has a forward part of $80\ \text{N}\cos 40^\circ = 61.3\ \text{N}$ and a lifting part of $80\ \text{N}\sin 40^\circ = 51.4\ \text{N}$. Notice that $61.3\ \text{N} + 51.4\ \text{N}$ is more than $80\ \text{N}$. Parts at right angles do not add as plain numbers.

Practise with new numbers

Try it in Python

The program adds the three walks of the worked example by components.

Remember these

• $A_x = A\cos\theta$ and $A_y = A\sin\theta$, when $\theta$ is measured from the $+x$ axis.
• $\vec{A} = A_x\hat{i} + A_y\hat{j}$, $A = \sqrt{A_x^2 + A_y^2}$, and the signs of the parts give the quarter.
• To add vectors, add their $x$ parts and their $y$ parts separately.

Check yourself

Try each one on your own first. If you are stuck, open a hint. When you have an answer, check it with Show the answer.

1. A vector of size $10$ makes $30^\circ$ with the $+x$ axis. What are its parts?

2. $\vec{A} = 3\hat{i} + 4\hat{j}$ and $\vec{B} = \hat{i} - 2\hat{j}$. Find $\vec{A} + \vec{B}$, its size, and $\vec{A} - \vec{B}$.

3. In which direction does $\vec{C} = -\hat{i} - \hat{j}$ point, as an angle from $+x$?

4. A boy pulls a sledge with a rope at $60^\circ$ above the ground, with a force of $50\ \text{N}$. How much of the pull moves the sledge forward?