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Vectors — Displacement or translation vectors

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Vectors — Displacement or Translation Vectors in 3 Dimensions

In three dimensions, a displacement vector describes how a point moves in space. It shows the change in the \(x\)-direction, the \(y\)-direction, and the \(z\)-direction.

In Cambridge A Level mathematics, 3D vectors are used to describe positions, translations, lines, and geometric relationships in space. A clear understanding of displacement vectors is essential for Year 13 vector work.

Key definitions and formulae

1. A translation vector in 3D

A translation vector in three dimensions is written as

\[ \begin{pmatrix} a\\ b\\ c \end{pmatrix}. \]

This means move \(a\) units in the \(x\)-direction, \(b\) units in the \(y\)-direction, and \(c\) units in the \(z\)-direction.

2. Translating a point

If the point \(P(x,y,z)\) is translated by the vector

\[ \begin{pmatrix} a\\ b\\ c \end{pmatrix}, \]

then its image is

\[ P'(x+a,\;y+b,\;z+c). \]

3. Displacement vector from one point to another

If \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\) are two points, then the displacement vector from \(A\) to \(B\) is

\[ \overrightarrow{AB}= \begin{pmatrix} x_2-x_1\\ y_2-y_1\\ z_2-z_1 \end{pmatrix}. \]

This is found by subtracting the coordinates of the starting point from the coordinates of the finishing point.

4. Position vectors

If the position vector of a point \(A\) is

\[ \mathbf{a}= \begin{pmatrix} x_1\\ y_1\\ z_1 \end{pmatrix} \quad \text{and} \quad \mathbf{b}= \begin{pmatrix} x_2\\ y_2\\ z_2 \end{pmatrix} \]

for point \(B\), then

\[ \overrightarrow{AB}=\mathbf{b}-\mathbf{a}. \]

Worked examples

Example 1: Translate a point in 3D

Translate the point \(A(2,-1,4)\) by the vector

\[ \begin{pmatrix} 3\\ 5\\ -2 \end{pmatrix}. \]

Solution

Add each component of the vector to the corresponding coordinate:

\[ A'=(2+3,\,-1+5,\;4-2)=(5,4,2). \]
The image of \(A\) is \((5,4,2)\).

Example 2: Find a displacement vector

Find the displacement vector from \(P(1,4,-2)\) to \(Q(7,-3,5)\).

Solution

Use finish minus start:

\[ \overrightarrow{PQ}= \begin{pmatrix} 7-1\\ -3-4\\ 5-(-2) \end{pmatrix} = \begin{pmatrix} 6\\ -7\\ 7 \end{pmatrix}. \]
The displacement vector is \[ \begin{pmatrix} 6\\ -7\\ 7 \end{pmatrix}. \]

Example 3: Use position vectors

The position vectors of points \(A\) and \(B\) are

\[ \mathbf{a}= \begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix}, \qquad \mathbf{b}= \begin{pmatrix} 5\\ 4\\ -2 \end{pmatrix}. \]

Find \(\overrightarrow{AB}\).

Solution

\[ \overrightarrow{AB}=\mathbf{b}-\mathbf{a} = \begin{pmatrix} 5\\ 4\\ -2 \end{pmatrix} - \begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix} = \begin{pmatrix} 3\\ 5\\ -5 \end{pmatrix}. \]
So, \[ \overrightarrow{AB}= \begin{pmatrix} 3\\ 5\\ -5 \end{pmatrix}. \]

Example 4: Find an unknown point

The point \(A(3,2,-1)\) is translated by the vector

\[ \begin{pmatrix} -4\\ 6\\ 3 \end{pmatrix} \]

to the point \(B\). Find the coordinates of \(B\).

Solution

\[ B=(3-4,\;2+6,\;-1+3)=(-1,8,2). \]
The coordinates of \(B\) are \((-1,8,2)\).

Standard method

When finding a displacement vector in 3D, always use:

\[ \text{displacement vector}=\text{final position}-\text{initial position}. \]

So if a point moves from \(A(x_1,y_1,z_1)\) to \(B(x_2,y_2,z_2)\), write

\[ \overrightarrow{AB}= \begin{pmatrix} x_2-x_1\\ y_2-y_1\\ z_2-z_1 \end{pmatrix}. \]

This method is used repeatedly in Year 13 vector questions, especially when working with lines and geometric problems in space.

Common mistakes and exam tips

Common mistakes

  • Subtracting in the wrong order when finding \(\overrightarrow{AB}\).
  • Forgetting the third coordinate.
  • Making sign errors when a coordinate is negative.
  • Confusing a point such as \((2,3,4)\) with a vector such as \(\begin{pmatrix}2\\3\\4\end{pmatrix}\).

Exam tips

  • Use the phrase finish minus start to avoid reversing the vector.
  • Write vectors clearly as column vectors.
  • Check each coordinate separately, especially the \(z\)-coordinate.
  • In longer questions, label position vectors and displacement vectors carefully to avoid confusion.

Summary

\[ P(x,y,z)\rightarrow P'(x+a,\;y+b,\;z+c) \] \[ \overrightarrow{AB}= \begin{pmatrix} x_2-x_1\\ y_2-y_1\\ z_2-z_1 \end{pmatrix} \] \[ \overrightarrow{AB}=\mathbf{b}-\mathbf{a} \]

A displacement or translation vector in 3D shows how far and in what direction a point moves in space.

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