The Dot Product: Quick Revision

A brief note on The Dot Product. Read it before a test. Try each question at the end before you open its answer.

Definition and rules

• Definition: $\vec a\cdot\vec b = \lvert\vec a\rvert\lvert\vec b\rvert\cos\theta$, with $0 \le \theta \le \pi$.
• Components: $\vec a\cdot\vec b = a_1b_1 + a_2b_2 + a_3b_3$.
• Sign: positive for an acute angle, zero for a right angle, negative for an obtuse angle.
• Rules: commutative, distributive, $\vec a\cdot\vec a = \lvert\vec a\rvert^2$, and $\lvert\vec a \pm \vec b\rvert^2 = \lvert\vec a\rvert^2 \pm 2\,\vec a\cdot\vec b + \lvert\vec b\rvert^2$.
• Limit: $\lvert\vec a\cdot\vec b\rvert \le \lvert\vec a\rvert\lvert\vec b\rvert$.

Angles and perpendicular vectors

• Angle: $\cos\theta = \dfrac{\vec a\cdot\vec b}{\lvert\vec a\rvert\lvert\vec b\rvert}$.
• Perpendicular: $\vec a\cdot\vec b = 0$ for non-zero vectors.
• Parallelogram: the diagonals are perpendicular for a rhombus and equal for a rectangle.
• Lines: use $\lvert\vec a\cdot\vec b\rvert$, so the angle is acute or right.

Projection and work

• Scalar projection of $\vec a$ on $\vec b$: $\dfrac{\vec a\cdot\vec b}{\lvert\vec b\rvert}$.
• Vector projection: $\dfrac{\vec a\cdot\vec b}{\lvert\vec b\rvert^2}\vec b$.
• Work: $W = \vec F\cdot\vec d$ for a constant force. Power: $P = \vec F\cdot\vec v$.

A worked example

Find the angle between $\vec a = 2\hat i + 2\hat j - \hat k$ and $\vec b = 6\hat i + 3\hat j + 2\hat k$, and the scalar projection of $\vec a$ on $\vec b$.

1. The dot product. $12 + 6 - 2 = 16$.
2. The magnitudes. $\lvert\vec a\rvert = 3$ and $\lvert\vec b\rvert = 7$.
3. The angle. $\cos\theta = \dfrac{16}{21}$, so $\theta \approx 40.37^\circ$.
4. The projection. $\dfrac{16}{7} \approx 2.29$.

Practice with new numbers

A program to try

The program checks the worked example.

Mistakes to avoid

• Giving a vector as the answer. The dot product is a number.
• Using the sine. The dot product uses the cosine.
• Dividing by $\lvert\vec b\rvert^2$ for the scalar projection.
• Forgetting the absolute value for the angle between lines.

Check yourself

1. Find $\vec a\cdot\vec b$ for $\vec a = 3\hat i - \hat j + 2\hat k$ and $\vec b = \hat i + 5\hat j + \hat k$.

2. If $\lvert\vec a\rvert = 4$, $\lvert\vec b\rvert = 6$ and the angle is $60^\circ$, find $\lvert\vec a - \vec b\rvert$.

3. Find the value of $p$ for which $2\hat i + p\hat j + \hat k$ is perpendicular to $\hat i - 3\hat j + 4\hat k$.

4. Find the scalar projection of $\hat i + \hat j + \hat k$ on $2\hat i - 2\hat j + \hat k$.

5. A force of $10$ N pulls at $60^\circ$ above the ground and moves a box $4$ m along the ground. How much work does it do?

Sources

• NCERT, Mathematics Part II for Class XII, Chapter 10, sections on the scalar product and the projection of a vector.
• OpenStax, Calculus Volume 3, Chapter 2, section on the dot product.