Adding, subtracting and multiplying by a scalar

A hiker walks from $\displaystyle A$ to $\displaystyle B$, and then from $\displaystyle B$ to $\displaystyle C$. The displacement of the two walks together is from $\displaystyle A$ to $\displaystyle C$. This is the triangle rule:

To add $\displaystyle \underline{u}+\underline{v}$, move $\displaystyle \underline{v}$ so that its tail is at the head of $\displaystyle \underline{u}$. The sum starts at the tail of $\displaystyle \underline{u}$ and ends at the head of $\displaystyle \underline{v}$. The same holds in a chain: $\displaystyle {\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}}={\overrightarrow{AD}}$.

When the two vectors leave the same point $\displaystyle A$, complete them to a parallelogram $\displaystyle ABCD$:

• the parallelogram rule: $\displaystyle {\underline{u}+\underline{v}}={\overrightarrow{AC}}$, the diagonal from $\displaystyle A$.
• subtracting: $\displaystyle {\underline{u}-\underline{v}}={\underline{u}+(-\underline{v})}$. In the parallelogram: $\displaystyle {\underline{u}-\underline{v}}={\overrightarrow{AB}-\overrightarrow{AD}}={\overrightarrow{DB}}$ – the other diagonal, from the head of $\displaystyle \underline{v}$ to the head of $\displaystyle \underline{u}$.

This gives a way to write any vector with vectors that leave one point $\displaystyle O$:

$\displaystyle t\cdot\underline{u}$, where $\displaystyle t$ is a number, is a vector parallel to $\displaystyle \underline{u}$:

• its length is $\displaystyle |t|\cdot|\underline{u}|$;
• when $\displaystyle t>0$ it points the same way as $\displaystyle \underline{u}$, and when $\displaystyle t<0$ the opposite way;
• $\displaystyle {0\cdot\underline{u}}={\underline{0}}$, and $\displaystyle {(-1)\cdot\underline{u}}={-\underline{u}}$.

For example, $\displaystyle 2\underline{u}$ is twice as long as $\displaystyle \underline{u}$, and $\displaystyle -\frac{1}{2}\underline{u}$ is half as long and points the other way.

For any vectors $\displaystyle \underline{u}$, $\displaystyle \underline{v}$, $\displaystyle \underline{w}$ and any numbers $\displaystyle s$, $\displaystyle t$:

• $\displaystyle {\underline{u}+\underline{v}}={\underline{v}+\underline{u}}$ (the commutative law);
• $\displaystyle {(\underline{u}+\underline{v})+\underline{w}}={\underline{u}+(\underline{v}+\underline{w})}$ (the associative law);
• $\displaystyle {\underline{u}+\underline{0}}={\underline{u}}$, and $\displaystyle {\underline{u}+(-\underline{u})}={\underline{0}}$;
• $\displaystyle {1\cdot\underline{u}}={\underline{u}}$, and $\displaystyle {s(t\underline{u})}={(st)\underline{u}}$;
• $\displaystyle {(s+t)\underline{u}}={s\underline{u}+t\underline{u}}$, and $\displaystyle {t(\underline{u}+\underline{v})}={t\underline{u}+t\underline{v}}$ (the distributive laws).

These are the laws numbers obey in adding and multiplying. So an expression like $\displaystyle 3(\underline{u}+2\underline{v})-2(\underline{u}-\underline{v})$ is simplified exactly like an algebraic expression: $\displaystyle \underline{u}+8\underline{v}$.

A plane flies east at 400 km/h. It enters an area where a wind blows to the north-west at 50 km/h.
What is the size of the plane’s velocity relative to the ground, and what is its direction?

The solution below adds the two velocity vectors by the triangle rule, and solves the triangle.

• The wind points $\displaystyle 135^{\circ}$ from east, so in the triangle the angle between the two vectors is $\displaystyle {180^{\circ}-135^{\circ}}={45^{\circ}}$.
• The law of cosines: $\displaystyle {|\underline{v}|^2}={400^2+50^2-2\cdot400\cdot50\cdot\cos45^{\circ}}$, so $\displaystyle |\underline{v}|\approx366.35$ km/h.
• The law of sines: $\displaystyle {\sin\beta}={\frac{50\sin45^{\circ}}{366.35}}$, so $\displaystyle \beta\approx5.54^{\circ}$ – the plane moves $\displaystyle 5.54^{\circ}$ north of east.

Simplify.

$\displaystyle ABCD$ is a parallelogram, $\displaystyle {\overrightarrow{AB}}={\underline{u}}$, $\displaystyle {\overrightarrow{AD}}={\underline{v}}$.
$\displaystyle M$ is the midpoint of $\displaystyle BC$, and $\displaystyle N$ the midpoint of $\displaystyle CD$.
Express with $\displaystyle \underline{u}$ and $\displaystyle \underline{v}$:

What does $\displaystyle \overrightarrow{MN}$ tell you about the segments $\displaystyle MN$ and $\displaystyle BD$?

A hiker walks 6 km east, and then 4 km in the direction $\displaystyle 60^{\circ}$ north of east.
How far is she from where she started? Round to two decimal places.

Two forces act on a ring: 30 newtons east and 40 newtons north.

What third force must act so that the sum of the forces is $\displaystyle \underline{0}$ and the ring stays in place? Describe its size and its direction.