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AI HL · AHL 4

Vectors

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AHL 4.4

Magnitude of a vector and unit vectors

AHL 4.4

Magnitude of a vector and unit vectors

Target: I can find the magnitude of a vector in 2D or 3D, in column or i, j, k form. (+3 more)

By the end of this section

Key points

Magnitude

  • The magnitude of \(\mathbf{a}=(a_1,a_2,a_3)\) is \(|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}\). It is Pythagoras in three dimensions.
  • Square every component first. A negative component gives a positive square.
  • The magnitude is a length, so it is never negative. The bars are the same symbol as the modulus.
  • Leave a surd exact unless the question asks for a decimal. In a context, give 3 significant figures.

Unit vectors

  • A unit vector has magnitude 1. The base vectors \(\mathbf{i},\mathbf{j},\mathbf{k}\) are unit vectors.
  • The unit vector in the direction of \(\mathbf{a}\) is \(\frac{1}{|\mathbf{a}|}\mathbf{a}\). Divide every component by the magnitude.
  • Two unit vectors are parallel to \(\mathbf{a}\): \(\frac{1}{|\mathbf{a}|}\mathbf{a}\) and \(-\frac{1}{|\mathbf{a}|}\mathbf{a}\).

A vector of a given magnitude

  • A vector of magnitude \(k\) in the direction of \(\mathbf{a}\) is \(k\times\frac{1}{|\mathbf{a}|}\mathbf{a}\).
  • For the opposite direction, use \(-k\) instead of \(k\).
  • Speed is the magnitude of the velocity vector. Force has a magnitude in newtons.

Explore: multiplying a vector by a scalar

Drag the tip of a and move the slider for k. The teal arrow is ka, drawn beside a. Say what it will do before you move the slider.

The tip of a snaps to whole numbers. Select it (Tab) and use the arrow keys to move it.

  1. Which values of k give a vector in the same direction as a? Which give the opposite direction? What happens at k = 0?
  2. Set k = −2. Is ka twice as long as a, or −2 times as long? A length can not be negative, so what does the minus sign tell you?
  3. Press the unit vector button for two different vectors a. What is the length of ka each time? Why?
  4. Choose k = 1/2. How do the components of ka compare with those of a? How does its length compare?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
The magnitude of a 3D vector

Find the magnitude of the vector \(\mathbf{a}=\begin{pmatrix}4 \\ -4 \\ 7\end{pmatrix}\).

Working
  1. Square each component

    \(|\mathbf{a}|=\sqrt{4^2+(-4)^2+7^2}\)

    A negative component gives a positive square.

  2. Add

    \(=\sqrt{81}\)

    16 + 16 + 49.

  3. Square root

    \(|\mathbf{a}|=9\)

    The magnitude is the length of the vector. It is never negative.

Video walkthrough coming
A unit vector parallel to a given vector

Find the two unit vectors parallel to \(\mathbf{b}=\begin{pmatrix}-6 \\ 0 \\ 8\end{pmatrix}\).

Working
  1. Magnitude

    \(|\mathbf{b}|=\sqrt{(-6)^2+0^2+8^2}=\sqrt{100}=10\)

    Find the length first.

  2. Divide by it

    \(\frac{1}{|\mathbf{b}|}\mathbf{b}=\frac{1}{10}\begin{pmatrix}-6 \\ 0 \\ 8\end{pmatrix}=\begin{pmatrix}- \frac{3}{5} \\ 0 \\ \frac{4}{5}\end{pmatrix}\)

    Dividing keeps the direction and makes the length 1.

  3. The other direction

    \(-\frac{1}{|\mathbf{b}|}\mathbf{b}=\begin{pmatrix}\frac{3}{5} \\ 0 \\ - \frac{4}{5}\end{pmatrix}\)

    There are two unit vectors parallel to b: one each way.

Video walkthrough coming
A velocity with a given speed

A drone flies in the direction of \(\begin{pmatrix}2 \\ 3 \\ 6\end{pmatrix}\) at a speed of 14 m/s. Find its velocity vector.

Working
  1. Unit vector

    \(|\mathbf{d}|=\sqrt{2^2+3^2+6^2}=\sqrt{49}=7,\quad \frac{1}{7}\begin{pmatrix}2 \\ 3 \\ 6\end{pmatrix}\)

    The direction, with length 1.

  2. Scale it

    \(14\times\frac{1}{7}\begin{pmatrix}2 \\ 3 \\ 6\end{pmatrix}=\begin{pmatrix}4 \\ 6 \\ 12\end{pmatrix}\)

    Multiplying a unit vector by 14 gives a vector of length 14.

  3. Check

    \(\sqrt{4^2+6^2+12^2}=14\)

    The speed is the magnitude of the velocity.

Video walkthrough coming