The dot product and its rules

What you will be able to do

• find the dot product of two vectors from the angle and from the components;
• use the sign of the dot product to tell an acute angle from a right angle and an obtuse angle;
• use the rules of the dot product to expand expressions such as $\lvert\vec a + \vec b\rvert^2$.

Pulling a trolley

You pull a trolley along a straight path with a rope. If the rope runs along the path, all of your pull helps. If you pull sideways, only part of your pull helps. If you pull at right angles to the path, none of it helps. We want one number that says how much two vectors point the same way. That number is the dot product.

The definition

Draw two non-zero vectors $\vec a$ and $\vec b$ from the same point. The angle between them, $\theta$, is the smaller turn from one to the other, so $0 \le \theta \le \pi$. The dot product is

The answer is a scalar, a plain number, so NCERT also calls it the scalar product. A-Level and IB books use the same name and may write $\mathbf a.\mathbf b$. If one of the vectors is zero, the dot product is $0$.

The sign of the dot product comes from $\cos\theta$. It is positive for an acute angle, zero for a right angle and negative for an obtuse angle.

The rules

For any vectors and any number $\lambda$:

• $\vec a\cdot\vec b = \vec b\cdot\vec a$ (it does not matter which comes first);
• $\vec a\cdot(\vec b + \vec c) = \vec a\cdot\vec b + \vec a\cdot\vec c$;
• $(\lambda\vec a)\cdot\vec b = \lambda(\vec a\cdot\vec b)$;
• $\vec a\cdot\vec a = \lvert\vec a\rvert^2$, because the angle between a vector and itself is $0$;
• $\hat i\cdot\hat i = \hat j\cdot\hat j = \hat k\cdot\hat k = 1$ and $\hat i\cdot\hat j = \hat j\cdot\hat k = \hat k\cdot\hat i = 0$, because the axes are at right angles.

The dot product from components

Write $\vec a = a_1\hat i + a_2\hat j + a_3\hat k$ and $\vec b = b_1\hat i + b_2\hat j + b_3\hat k$. Multiply out with the rules. The nine products give only three terms that are not zero, so

Equating the two forms gives the angle:

The square of a sum

Use $\vec a\cdot\vec a = \lvert\vec a\rvert^2$ and the rules:

In the same way, $\lvert\vec a - \vec b\rvert^2 = \lvert\vec a\rvert^2 - 2\,\vec a\cdot\vec b + \lvert\vec b\rvert^2$. This is the cosine rule of a triangle written with vectors.

Because $\lvert\cos\theta\rvert \le 1$, we always have $\lvert\vec a\cdot\vec b\rvert \le \lvert\vec a\rvert\,\lvert\vec b\rvert$. A-Level Further Maths and IB books call this the Cauchy-Schwarz inequality.

A worked example

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

1. The dot product. $3(1) + (-1)(4) + 2(-2) = 3 - 4 - 4 = -5$.
2. The sign. It is negative, so the angle is obtuse. Here $\lvert\vec a\rvert = \sqrt{14}$ and $\lvert\vec b\rvert = \sqrt{21}$, so $\cos\theta = \dfrac{-5}{\sqrt{294}} \approx -0.292$ and $\theta \approx 106.95^\circ$.
3. A check with the square of a sum. $\vec a + \vec b = 4\hat i + 3\hat j$ has $\lvert\vec a + \vec b\rvert^2 = 25$. Also $14 + 21 + 2(-5) = 25$.

A second example. Two vectors have $\lvert\vec a\rvert = 2$, $\lvert\vec b\rvert = 3$ and $\vec a\cdot\vec b = -3$. Then $\lvert\vec a + \vec b\rvert^2 = 4 + 9 - 6 = 7$ and $\lvert\vec a - \vec b\rvert^2 = 4 + 9 + 6 = 19$.

Practise with new numbers

Try it in Python

The program finds the dot product of the worked example in two ways, then checks the square of a sum and the Cauchy-Schwarz limit.

Common mistakes

• Giving a vector as the answer. The dot product is a number.
• Writing $(\vec a\cdot\vec b)\cdot\vec c$. The dot product needs two vectors. A number dotted with a vector means nothing.
• Thinking $\vec a\cdot\vec b = 0$ means that one vector is zero. It also happens when the vectors are perpendicular.
• Cancelling. From $\vec a\cdot\vec b = \vec a\cdot\vec c$ you cannot conclude $\vec b = \vec c$.
• Using $\sin\theta$. The dot product uses the cosine. The sine belongs to the cross product, in the next course.

Remember these

• $\vec a\cdot\vec b = \lvert\vec a\rvert\lvert\vec b\rvert\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$.
• It is positive for an acute angle, zero for a right angle and negative for an obtuse angle.
• $\vec a\cdot\vec a = \lvert\vec a\rvert^2$ and $\lvert\vec a \pm \vec b\rvert^2 = \lvert\vec a\rvert^2 \pm 2\,\vec a\cdot\vec b + \lvert\vec b\rvert^2$.
• $\lvert\vec a\cdot\vec b\rvert \le \lvert\vec a\rvert\lvert\vec b\rvert$.

Check yourself

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

1. If $\lvert\vec a\rvert = 4$, $\lvert\vec b\rvert = 3$ and the angle between them is $60^\circ$, find $\vec a\cdot\vec b$.

2. Find $\vec a\cdot\vec b$ for $\vec a = 2\hat i + 3\hat j - \hat k$ and $\vec b = \hat i - 2\hat j + 4\hat k$. Is the angle acute or obtuse?

3. For two vectors with $\lvert\vec a\rvert = 5$ and $\lvert\vec b\rvert = 2$, what is the largest value that $\vec a\cdot\vec b$ can have?

4. Does $(\vec a\cdot\vec b)\cdot\vec c$ make sense?

5. If $\vec a\cdot\vec b = \vec a\cdot\vec c$ and $\vec a \ne \vec 0$, must $\vec b = \vec c$? Give an example.

References

• NCERT, Mathematics Part II for Class XII, Chapter 10, section on the scalar (or dot) product of vectors.
• OpenStax, Calculus Volume 3, Chapter 2, section on the dot product.