Vectors, components and direction cosines

What you will be able to do

• tell a vector from a scalar, and write a vector with its components;
• find the magnitude, the unit vector and the direction cosines of a vector;
• use the direction angles and the rule $l^2 + m^2 + n^2 = 1$.

Where is the ship?

A ship leaves a harbour. It sails $6$ km north and then $8$ km east. The distance it sailed is $14$ km. That is one number, and it does not tell you where the ship is. The harbour master wants to know how far the ship is from the harbour, and in which direction. The straight-line distance is $10$ km, and it points to the north-east side. So we need a quantity that has both a size and a direction. We call it a vector.

Scalars and vectors

A scalar has a size only. Mass, time, speed, temperature and energy are scalars. A vector has a size and a direction. Displacement, velocity, acceleration and force are vectors.

We draw a vector as an arrow. The length of the arrow is the magnitude, and the arrowhead shows the direction. The arrow $\overrightarrow{AB}$ starts at the initial point $A$ and ends at the terminal point $B$. We write $\vec a$ for a vector and $\lvert\vec a\rvert$ for its magnitude. A-Level and IB books print a vector in bold, $\mathbf{a}$, and write it by hand with a line under the letter. The magnitude is also called the modulus.

Some names you will meet in the Indian syllabus:

• A zero vector has magnitude $0$ and no direction.
• A unit vector has magnitude $1$. We write $\hat a$, read "a hat".
• Equal vectors have the same magnitude and the same direction. Where you draw them does not matter.
• The negative of $\vec a$ has the same magnitude and the opposite direction. We write $-\vec a$.
• Collinear vectors lie along the same line, or along parallel lines. They may point the same way or opposite ways.

Components

Take three axes at right angles to each other, meeting at the origin $O$. Let $\hat i$, $\hat j$ and $\hat k$ be unit vectors along the $x$, $y$ and $z$ axes. A-Level and IB write them as $\mathbf i$, $\mathbf j$, $\mathbf k$, and often write the vector as a column of numbers. The three axes in this book are right-handed. In a picture, the $z$-axis points up. Turn the $x$-axis towards the $y$-axis, and a right-handed screw moves along the $z$-axis.

The point $P(x, y, z)$ has the position vector $\overrightarrow{OP}$ from the origin. We reach $P$ in three steps: $x$ along the $x$-axis, then $y$ parallel to the $y$-axis, then $z$ upwards.

The numbers $x$, $y$ and $z$ are the components of the vector.

Magnitude

Use Pythagoras twice. The diagonal of the floor is $\sqrt{x^2 + y^2}$. The vector $\overrightarrow{OP}$ is the diagonal of a vertical rectangle with that floor diagonal and the height $z$ as its sides. So

For $P(2, 3, 6)$ the floor diagonal is $\sqrt{13}$, and $\lvert\overrightarrow{OP}\rvert = \sqrt{13 + 36} = \sqrt{49} = 7$.

Unit vector

To get a vector of size $1$ in the direction of $\vec a$, divide $\vec a$ by its own magnitude:

For $\vec a = 2\hat i + 3\hat j + 6\hat k$ we get $\hat a = \dfrac{2}{7}\hat i + \dfrac{3}{7}\hat j + \dfrac{6}{7}\hat k$. A vector of magnitude $m$ in the same direction is $m\hat a$.

Direction cosines

The vector $\vec r$ makes the angles $\alpha$, $\beta$ and $\gamma$ with the positive $x$, $y$ and $z$ axes. These are the direction angles, and each lies between $0$ and $\pi$.

In the right-angled triangle of the figure, the component along an axis equals $\lvert\vec r\rvert$ times the cosine of the angle with that axis. So $x = \lvert\vec r\rvert\cos\alpha$. The three numbers

are the direction cosines. They are the components of the unit vector, $\hat r = l\hat i + m\hat j + n\hat k$. Since $\hat r$ has magnitude $1$,

Any three numbers proportional to $l$, $m$, $n$ are called direction ratios. For example, the direction ratios $6, -6, 3$ are the same as $2, -2, 1$. Divide by $\sqrt{4 + 4 + 1} = 3$ to get the direction cosines $\dfrac{2}{3}, -\dfrac{2}{3}, \dfrac{1}{3}$.

A vector has one set of direction cosines. A line has two sets, because it can be followed in either direction, and the second set has all three signs changed. A negative cosine means an obtuse angle. A-Level and IB courses usually find these angles with the dot product, which is the next course of this chapter.

A worked example

Find the magnitude, the unit vector and the direction angles of $\vec a = \hat i - 2\hat j + 2\hat k$.

1. Magnitude. $\sqrt{1 + 4 + 4} = 3$.
2. Unit vector. $\hat a = \dfrac{1}{3}\hat i - \dfrac{2}{3}\hat j + \dfrac{2}{3}\hat k$. The direction cosines are $\dfrac{1}{3}$, $-\dfrac{2}{3}$ and $\dfrac{2}{3}$. Their squares add to $\dfrac{1 + 4 + 4}{9} = 1$.
3. Angles. $\alpha = \cos^{-1}\dfrac{1}{3} \approx 70.53^\circ$. $\beta = \cos^{-1}\left(-\dfrac{2}{3}\right) \approx 131.81^\circ$, which is obtuse because the $y$-component is negative. $\gamma = \cos^{-1}\dfrac{2}{3} \approx 48.19^\circ$.

Practise with new numbers

Try it in Python

The program finds the magnitude, the unit vector and the direction angles of $2\hat i + 3\hat j + 6\hat k$ and checks that $l^2 + m^2 + n^2 = 1$.

Common mistakes

• Forgetting the square root. The magnitude of $3\hat i + 4\hat j$ is $5$, not $25$.
• Adding the components. The size of $2\hat i + 3\hat j + 6\hat k$ is $7$, not $11$.
• Mixing up a unit vector and a vector of length one in any direction. The unit vector must point along the given vector.
• Dropping a sign. A negative component gives a negative direction cosine and an obtuse angle.
• Thinking the three angles add up to a fixed number. It is the squares of the cosines that add up to $1$.

Remember these

• A vector has a magnitude and a direction. A scalar has a magnitude only.
• $\overrightarrow{OP} = x\hat i + y\hat j + z\hat k$ and $\lvert\overrightarrow{OP}\rvert = \sqrt{x^2 + y^2 + z^2}$.
• $\hat a = \dfrac{\vec a}{\lvert\vec a\rvert}$, and its components are the direction cosines $l, m, n$.
• $l^2 + m^2 + n^2 = 1$.

Check yourself

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

1. Which of these are vectors: mass, velocity, temperature, force, speed?

2. Find the magnitude of $6\hat i + 2\hat j - 3\hat k$.

3. Find the unit vector along $8\hat i - \hat j - 4\hat k$.

4. A line makes $60^\circ$ with the $x$-axis and $60^\circ$ with the $y$-axis. What angle can it make with the $z$-axis?

5. Can $0.6$, $0.8$ and $0.2$ be the direction cosines of a line?

References

• NCERT, Mathematics Part II for Class XII, Chapter 10, sections on some basic concepts and on direction cosines.
• OpenStax, Calculus Volume 3, Chapter 2, sections on vectors in the plane and vectors in three dimensions.