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AI HL · AHL 4

Vectors

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AHL 4.3

Scalar multiplication and parallel vectors

AHL 4.3

Scalar multiplication and parallel vectors

Target: I can multiply a vector by a scalar and find a linear combination such as 3a − 2b of column vectors. (+3 more)

By the end of this section

Key points

Multiplying by a scalar

  • Multiply every component by the scalar: \(2\begin{pmatrix}3\\-5\\0\end{pmatrix}=\begin{pmatrix}6\\-10\\0\end{pmatrix}\).
  • The direction stays the same and the length is multiplied by the scalar.
  • A negative scalar reverses the direction. \(-\mathbf{a}\) is \(\mathbf{a}\) turned round.
  • In a combination such as \(3\mathbf{a}-2\mathbf{b}\), multiply first and then add or subtract the components.

Parallel vectors

  • If \(\mathbf{a}\) and \(\mathbf{b}\) are parallel then \(\mathbf{b}=t\mathbf{a}\) for some scalar \(t\). The scalar can be positive or negative.
  • Positive t: the same direction. Negative t: the opposite direction.
  • Equal vectors have equal components, so \(\mathbf{b}=t\mathbf{a}\) gives one equation per component. Use the component with no unknown to find t first.
  • To show two vectors are parallel, find t from one component and check that it works for all of them.

Vectors in a parallelogram

  • Opposite sides of a parallelogram are equal and parallel: \(\overrightarrow{DC}=\overrightarrow{AB}\) and \(\overrightarrow{BC}=\overrightarrow{AD}\).
  • A midpoint gives a half: if M is the midpoint of AD then \(\overrightarrow{AM}=\frac{1}{2}\overrightarrow{AD}\).
  • For an extended side, say how long the extra piece is. If CN = 2BC then \(\overrightarrow{CN}=2\overrightarrow{BC}\), in the same direction.
  • For a vector between two points that are not joined by one side, go by a path along sides you know. Travelling against a vector means subtract it.

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.
A linear combination

Given \(\mathbf{a}=\begin{pmatrix}5\\-2\\3\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}-4\\1\\6\end{pmatrix}\), find \(3\mathbf{a}-2\mathbf{b}\).

Working
  1. Multiply first

    \(3\mathbf{a}-2\mathbf{b}=3\begin{pmatrix}5\\-2\\3\end{pmatrix}-2\begin{pmatrix}-4\\1\\6\end{pmatrix}\)

    Write each scalar in front of its vector.

  2. Scale each vector

    \(=\begin{pmatrix}15\\-6\\9\end{pmatrix}-\begin{pmatrix}-8\\2\\12\end{pmatrix}\)

    Multiply every component. Do the multiplying before any adding or subtracting.

  3. Subtract

    \(=\begin{pmatrix}23\\-8\\-3\end{pmatrix}\)

    Now subtract the second column from the first, row by row.

Video walkthrough coming
Parallelogram with a midpoint and an extended side

PQRS is a parallelogram with \(\overrightarrow{PQ}=\mathbf{a}\) and \(\overrightarrow{PS}=\mathbf{b}\). X is the midpoint of QR. Y is on the line PQ extended beyond Q, with QY = 2PQ. Find, in terms of \(\mathbf{a}\) and \(\mathbf{b}\): (a) \(\overrightarrow{QX}\) and \(\overrightarrow{QY}\); (b) \(\overrightarrow{XY}\).

abPQRSXY
Working
  1. Parallel sides

    \(\overrightarrow{QR}=\overrightarrow{PS}=\mathbf{b}\)

    In a parallelogram opposite sides are equal and parallel, so QR is b.

  2. A half

    \(\overrightarrow{QX}=\frac{1}{2}\overrightarrow{QR}=\frac{1}{2}\mathbf{b}\)

    X is the midpoint, so QX is half of QR, in the same direction.

  3. A multiple of a

    \(\overrightarrow{QY}=2\overrightarrow{PQ}=2\mathbf{a}\)

    QY has the same direction as PQ and twice the length.

  4. Go by a path

    \(\overrightarrow{XY}=\overrightarrow{XQ}+\overrightarrow{QY}=-\frac{1}{2}\mathbf{b}+2\mathbf{a}\)

    From X to Y by way of Q. XQ goes against QX, so it is the negative.

  5. Tidy

    \(\overrightarrow{XY}=2\mathbf{a}-\frac{1}{2}\mathbf{b}\)

    Write the a term first.

Video walkthrough coming
Parallel vectors

Given \(\mathbf{a}=\begin{pmatrix}4\\-6\\10\end{pmatrix}\) and \(\mathbf{c}=\begin{pmatrix}-6\\p\\q\end{pmatrix}\), find the values of \(p\) and \(q\) so that \(\mathbf{c}\) is parallel to \(\mathbf{a}\).

Working
  1. Write c as a multiple of a

    \(\mathbf{c}=t\mathbf{a}:\quad\begin{pmatrix}-6\\p\\q\end{pmatrix}=t\begin{pmatrix}4\\-6\\10\end{pmatrix}=\begin{pmatrix}4t\\-6t\\10t\end{pmatrix}\)

    Parallel vectors are scalar multiples. t can be positive or negative.

  2. Use the known component

    \(4t=-6\Rightarrow t=-\frac{3}{2}\)

    Equal vectors have equal components. The first component is the only one with no unknown.

  3. Find p and q

    \(p=-6t=9,\quad q=10t=-15\)

    Put t back into the other two components. t is negative, so c points the opposite way to a.

Video walkthrough coming