Adding and scaling vectors

What you will be able to do

• add and subtract vectors by the triangle law, the parallelogram law and by components;
• multiply a vector by a number, and test whether two vectors are collinear;
• find the vector from one point to another, and its magnitude.

A boat on a river

A boat heads straight across a river, north, at $4$ m/s. The river flows east at $3$ m/s. The boat does not go north, and it does not go east. It goes in a direction between the two. Both movements happen at once, so the real velocity is the sum of the two velocity vectors. We now learn how to add vectors.

The triangle law and the parallelogram law

Put the tail of $\vec b$ at the head of $\vec a$. The vector from the tail of $\vec a$ to the head of $\vec b$ is the sum $\vec a + \vec b$. This is the triangle law. If we draw $\vec a$ and $\vec b$ from the same point, the diagonal of the parallelogram they make is also $\vec a + \vec b$. This is the parallelogram law.

Adding vectors is commutative, $\vec a + \vec b = \vec b + \vec a$, and associative, $(\vec a + \vec b) + \vec c = \vec a + (\vec b + \vec c)$. The zero vector does nothing: $\vec a + \vec 0 = \vec a$. The difference $\vec a - \vec b$ means $\vec a + (-\vec b)$.

Adding in components

If $\vec a = a_1\hat i + a_2\hat j + a_3\hat k$ and $\vec b = b_1\hat i + b_2\hat j + b_3\hat k$, then

We add the matching components. We subtract in the same way.

Multiplying by a number

For a real number $\lambda$, the vector $\lambda\vec a$ has magnitude $\lvert\lambda\rvert\,\lvert\vec a\rvert$. It points the same way as $\vec a$ if $\lambda > 0$ and the opposite way if $\lambda < 0$. In components, $\lambda\vec a = \lambda a_1\hat i + \lambda a_2\hat j + \lambda a_3\hat k$.

Two non-zero vectors are collinear, or parallel, when one is a number times the other, $\vec a = \lambda\vec b$. Their components are then in the same ratio. The vectors $2\hat i - 4\hat j + 6\hat k$ and $-\hat i + 2\hat j - 3\hat k$ are collinear, because the first is $-2$ times the second. They point in opposite directions.

The size of a sum

The size of $\vec a + \vec b$ depends on the angle between the vectors. If they are at right angles, Pythagoras gives $\lvert\vec a + \vec b\rvert^2 = \lvert\vec a\rvert^2 + \lvert\vec b\rvert^2$. In general, a side of a triangle is never longer than the other two sides together, so

This is the triangle inequality. The two sides are equal only when the vectors point the same way.

The vector from one point to another

Let $A$ and $B$ have position vectors $\vec a$ and $\vec b$. Going from $A$ to $B$ means going back from $A$ to $O$ and then on to $B$, so

Remember it as "head minus tail". If $A(1, 2, 3)$ and $B(4, 6, 15)$, then $\overrightarrow{AB} = 3\hat i + 4\hat j + 12\hat k$, and the distance between the points is $\lvert\overrightarrow{AB}\rvert = \sqrt{9 + 16 + 144} = 13$.

If you walk round a closed path, the displacements add up to the zero vector. For a triangle, $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \vec 0$.

A worked example

Let $\vec a = \hat i + 2\hat j + 3\hat k$ and $\vec b = 3\hat i - \hat j + 2\hat k$. Find $\vec a + \vec b$, $\vec a - \vec b$ and $3\vec a - 2\vec b$.

1. The sum. $(1 + 3)\hat i + (2 - 1)\hat j + (3 + 2)\hat k = 4\hat i + \hat j + 5\hat k$. Its size is $\sqrt{42} \approx 6.48$.
2. The difference. $-2\hat i + 3\hat j + \hat k$. Its size is $\sqrt{14} \approx 3.74$.
3. A combination. $3\vec a = 3\hat i + 6\hat j + 9\hat k$ and $2\vec b = 6\hat i - 2\hat j + 4\hat k$, so $3\vec a - 2\vec b = -3\hat i + 8\hat j + 5\hat k$.

Now the boat. The velocity over the ground is $3\hat i + 4\hat j$ in metres per second, east and north. Its size is $\sqrt{9 + 16} = 5$ m/s. The direction is $\tan^{-1}\dfrac{4}{3} \approx 53.13^\circ$ north of east.

Practise with new numbers

Try it in Python

The program adds and subtracts the vectors of the worked example, checks the triangle inequality and adds a closed path.

Common mistakes

• Adding the magnitudes. $\lvert\vec a + \vec b\rvert$ is not $\lvert\vec a\rvert + \lvert\vec b\rvert$, except when the vectors point the same way.
• Tail minus head. The vector from $A$ to $B$ is $B$ minus $A$, head minus tail.
• Forgetting that a number times a vector is a vector. $-3\vec a$ is three times as long and points the other way.
• Joining the wrong ends. In the triangle law the tail of the second vector goes at the head of the first.

Remember these

• Add and subtract vectors by components. The triangle law and the parallelogram law give the same result.
• $\lvert\vec a + \vec b\rvert \le \lvert\vec a\rvert + \lvert\vec b\rvert$.
• $\vec a$ and $\vec b$ are collinear when $\vec a = \lambda\vec b$.
• $\overrightarrow{AB} = \vec b - \vec a$.

Check yourself

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

1. Let $\vec a = 2\hat i - \hat j + 3\hat k$ and $\vec b = \hat i + 4\hat j - \hat k$. Find $\lvert\vec a + \vec b\rvert$.

2. With the same vectors, find $2\vec a - 3\vec b$.

3. Are $2\hat i - 3\hat j + 5\hat k$ and $-6\hat i + 9\hat j - 15\hat k$ collinear? Do they point the same way?

4. Find $\overrightarrow{AB}$ and its magnitude for $A(1, -2, 3)$ and $B(-3, 2, 1)$.

5. Two vectors have magnitudes $3$ and $4$. Can their sum have magnitude $8$? What are the smallest and largest possible magnitudes?

References

• NCERT, Mathematics Part II for Class XII, Chapter 10, sections on addition of vectors and on multiplication of a vector by a scalar.
• OpenStax, University Physics Volume 1, Chapter 2, sections on coordinate systems and components and on algebra of vectors.