Vectors in a Plane: Quick Revision

A brief note on Vectors in a Plane. Read it the evening before a test, or when a vector problem has you stuck. Try each question at the end before you open its answer.

Scalars and vectors

• A scalar has a size only. Mass, time, distance, speed and energy are scalars.
• A vector has a size and a direction. Displacement, velocity, acceleration and force are vectors.
• Two vectors are equal when they have the same size and the same direction. Where they are drawn does not matter.
• Multiplying a vector by a number $k$ multiplies its size by $|k|$. A negative $k$ also turns it round.

The formulas at a glance

• The size of $\vec{A} + \vec{B}$: $\sqrt{A^2 + B^2 + 2AB\cos\theta}$
• The size of $\vec{A} - \vec{B}$: $\sqrt{A^2 + B^2 - 2AB\cos\theta}$
• The parts of $\vec{A}$: $A\cos\theta$ and $A\sin\theta$
• The size from the parts: $\sqrt{A_x^2 + A_y^2}$
• The direction from the parts: $\tan\theta = A_y / A_x$
• The unit vector along $\vec{A}$: $\hat{A} = \vec{A} / A$

In the first two rows, $\theta$ is the angle between $\vec{A}$ and $\vec{B}$ drawn tail to tail. For the parts, $\theta$ is measured from the $+x$ axis.

The sum lies between two limits

• Same direction ($\theta = 0^\circ$): the sizes add, $R = A + B$.
• Opposite directions ($\theta = 180^\circ$): $R = |A - B|$.
• At right angles ($\theta = 90^\circ$): $R = \sqrt{A^2 + B^2}$.

Adding by parts: the method that always works

1. Split every vector into its $x$ and $y$ parts.
2. Add all the $x$ parts. Add all the $y$ parts.
3. Join the two totals: the size by Pythagoras, the direction from $\tan\theta$.

An example. A boy walks $7\ \text{m}$ at $40^\circ$ from the $+x$ axis, then $4\ \text{m}$ at $160^\circ$.

So $R = \sqrt{(1.60\ \text{m})^2 + (5.87\ \text{m})^2} = 6.08\ \text{m}$, at $\tan^{-1}(5.87\ \text{m}/1.60\ \text{m}) = 74.7^\circ$ from the $+x$ axis.

Check it another way. The two walks are $160^\circ - 40^\circ = 120^\circ$ apart. So $R^2 = (7\ \text{m})^2 + (4\ \text{m})^2 + 2 \times 7\ \text{m} \times 4\ \text{m} \times \cos 120^\circ = 37\ \text{m}^2$, and $R = 6.08\ \text{m}$ again.

Unit vectors

$\hat{i}$ and $\hat{j}$ have size $1$ and point along $+x$ and $+y$. Any vector in the plane is $\vec{A} = A_x\hat{i} + A_y\hat{j}$.

For $\vec{P} = 9\hat{i} - 12\hat{j}$, the size is $\sqrt{81 + 144} = 15$. So $\hat{P} = 0.6\hat{i} - 0.8\hat{j}$. Check: $0.6^2 + 0.8^2 = 1$, as a unit vector must.

Practice with new numbers

A program to try

This program adds any list of vectors by their parts, exactly as above.

Mistakes to avoid

• Adding the sizes. $3\ \text{N}$ and $4\ \text{N}$ make $7\ \text{N}$ only when they point the same way. At right angles they make $5\ \text{N}$.
• The wrong angle in the formula. In $R^2 = A^2 + B^2 + 2AB\cos\theta$, $\theta$ is the angle between the vectors drawn tail to tail. It is not the angle inside the triangle.
• Cosine and sine swapped. The part next to the angle takes $\cos$. The part opposite the angle takes $\sin$.
• The wrong quarter. $\tan\theta = A_y/A_x$ gives the same number for $\vec{A}$ and $-\vec{A}$. Look at the signs of the parts to see where the vector points.

Check yourself

1. Two forces of $4\ \text{N}$ and $10\ \text{N}$ act on a point. Which of these can be their resultant: $5\ \text{N}$, $7\ \text{N}$, $13\ \text{N}$, $15\ \text{N}$?

2. A vector of size $25$ has an $x$ part of $7$. What is its $y$ part (take it as positive), and what angle does the vector make with the $x$ axis?

3. Two vectors of the same size $F$ add up to a vector that also has the size $F$. What is the angle between them?

4. $\vec{A} = 3\hat{i} + 5\hat{j}$ and $\vec{B} = 6\hat{i} - 2\hat{j}$. Find $\vec{A} - \vec{B}$ and its size.

5. Can two vectors of different sizes add up to zero? Can three?