Projections and work done

What you will be able to do

• find the scalar and the vector projection of one vector on another;
• split a vector into a part along a given direction and a part at right angles to it;
• find the work done by a constant force, and use the dot product for a closest approach.

A shadow

Hold a stick in the sunshine at noon. Its shadow on the ground is shorter than the stick. The shadow is the projection of the stick on the ground. Vectors have shadows too. The shadow of $\vec a$ on the line of $\vec b$ shows how much of $\vec a$ points along $\vec b$.

Projection

Let $\theta$ be the angle between $\vec a$ and $\vec b$. The scalar projection of $\vec a$ on $\vec b$ is the signed length of the shadow:

It is negative when the angle is obtuse. The vector projection is that length along $\hat b$:

What is left of $\vec a$ is at right angles to $\vec b$:

By Pythagoras, $\lvert\vec a\rvert^2 = (\text{scalar projection})^2 + \lvert\vec a_{\perp}\rvert^2$. The size of $\vec a_{\perp}$ is also the shortest distance from the tip of $\vec a$ to the line through the origin in the direction of $\vec b$. A-Level books call the scalar projection the component of $\vec a$ in the direction of $\vec b$.

Work done

A constant force $\vec F$ moves an object through the displacement $\vec d$. Only the part of the force along the displacement does work, so

The unit is the joule when $F$ is in newtons and $d$ is in metres. The work is negative when the force opposes the motion, as friction does. It is zero when the force is at right angles to the motion, for example when a porter carries a bag along a level corridor with an upward force.

If the force is constant, the work depends only on the total displacement, because $\vec F\cdot(\vec d_1 + \vec d_2) = \vec F\cdot\vec d_1 + \vec F\cdot\vec d_2$. Power is $P = \vec F\cdot\vec v$, in watts.

A worked example

Let $\vec a = 4\hat i - 3\hat j + 5\hat k$ and $\vec b = \hat i + 2\hat j + 2\hat k$.

1. The numbers. $\vec a\cdot\vec b = 4 - 6 + 10 = 8$, and $\lvert\vec b\rvert = 3$.
2. The scalar projection. $\dfrac{8}{3} \approx 2.67$.
3. The vector projection. $\dfrac{8}{9}(\hat i + 2\hat j + 2\hat k) = \dfrac{8}{9}\hat i + \dfrac{16}{9}\hat j + \dfrac{16}{9}\hat k$.
4. The part across. $\vec a_{\perp} = \dfrac{28}{9}\hat i - \dfrac{43}{9}\hat j + \dfrac{29}{9}\hat k$. Its dot product with $\vec b$ is $\dfrac{28 - 86 + 58}{9} = 0$.
5. Pythagoras. $\lvert\vec a\rvert^2 = 50$. Also $\left(\dfrac{8}{3}\right)^2 + \lvert\vec a_{\perp}\rvert^2 = \dfrac{64}{9} + \dfrac{386}{9} = 50$.

A block of weight $200$ N slides $4$ m down a smooth slope that makes $30^\circ$ with the horizontal. Take the weight as $(0, -200)$ and the displacement as $4(\cos 30^\circ, -\sin 30^\circ)$. Then $W = 200 \times 4 \times \sin 30^\circ = 400$ J. Only the part of the weight along the slope does work.

Practise with new numbers

Try it in Python

The program splits $\vec a$ into a part along $\vec b$ and a part across it, for the worked example, and checks Pythagoras.

Common mistakes

• Dividing by $\lvert\vec b\rvert^2$ for the scalar projection. That is for the vector projection, which is then multiplied by $\vec b$.
• Mixing up the two projections. The projection of $\vec a$ on $\vec b$ is not the projection of $\vec b$ on $\vec a$.
• Forgetting the sign. An obtuse angle gives a negative scalar projection.
• Writing $W = Fd$ without the cosine.
• Adding the work of each leg of a path as if it were always positive. The work of a constant force is $\vec F\cdot$ (net displacement).

Remember these

• Scalar projection of $\vec a$ on $\vec b$: $\dfrac{\vec a\cdot\vec b}{\lvert\vec b\rvert}$. Vector projection: $\dfrac{\vec a\cdot\vec b}{\lvert\vec b\rvert^2}\vec b$.
• $\vec a = \text{projection} + \vec a_{\perp}$, with $\vec a_{\perp}\cdot\vec b = 0$.
• $W = \vec F\cdot\vec d$ for a constant force, and $P = \vec F\cdot\vec v$.

Check yourself

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

1. Find the scalar projection of $\vec a = \hat i + 2\hat j + 3\hat k$ on $\vec b = 2\hat i - \hat j + 2\hat k$.

2. Find the vector projection in question 1.

3. A force $(4, -3, 0)$ N moves an object from the origin to $(5, 2, 0)$ m. Find the work done.

4. A porter carries a suitcase along a level corridor at constant speed. How much work does the porter's upward force do on the suitcase?

5. Friction of $6$ N acts against a box that slides $2$ m. How much work does friction do?

References

• NCERT, Mathematics Part II for Class XII, Chapter 10, section on the projection of a vector on a line.
• OpenStax, University Physics Volume 1, Chapter 2, section on products of vectors, and Chapter 7, section on work.