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Worked solution

Find a position vector, its modulus and scaled directions

A point AA has coordinates (1,2,3)(1,2,3). Find (i) OA→\overrightarrow{OA}, (ii) ∣OA→∣\lvert\overrightarrow{OA}\rvert, (iii) the unit direction vector in the direction of OA→\overrightarrow{OA}, and (iv) a vector with magnitude 4 units in that direction.

CJC · Vectors Basic · Example 1

(i) Position vector

Answer

OA→=i+2j+3k=(123)\overrightarrow{OA}=\mathbf i+2\mathbf j+3\mathbf k=\begin{pmatrix}1\\2\\3\end{pmatrix}.

Read the three coordinates

OA→=i+2j+3k=(123)\overrightarrow{OA}=\mathbf i+2\mathbf j+3\mathbf k=\begin{pmatrix}1\\2\\3\end{pmatrix}

The coefficients are the displacements in the three orthonormal basis directions.

How did this part go?

(ii) Modulus

Answer

14\sqrt{14}.

Apply the three-dimensional distance formula

∣OA→∣=12+22+32=14\lvert\overrightarrow{OA}\rvert=\sqrt{1^2+2^2+3^2}=\sqrt{14}

The source magnitude bars and radical are essential; this is a scalar length.

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(iii) Unit direction

Answer

114(123)=1414(123)\dfrac1{\sqrt{14}}\begin{pmatrix}1\\2\\3\end{pmatrix}=\dfrac{\sqrt{14}}{14}\begin{pmatrix}1\\2\\3\end{pmatrix}.

Divide by the nonzero magnitude

OA→∣OA→∣=114(123)=1414(123)\frac{\overrightarrow{OA}}{\lvert\overrightarrow{OA}\rvert}=\frac1{\sqrt{14}}\begin{pmatrix}1\\2\\3\end{pmatrix}=\frac{\sqrt{14}}{14}\begin{pmatrix}1\\2\\3\end{pmatrix}

This retains both source forms. The origin would have no defined unit direction.

How did this part go?

(iv) Magnitude 4 in the same direction

Answer

2147(123)\dfrac{2\sqrt{14}}7\begin{pmatrix}1\\2\\3\end{pmatrix}.

Scale the unit vector by 4

4OA→∣OA→∣=4(1414)(123)=2147(123)4\frac{\overrightarrow{OA}}{\lvert\overrightarrow{OA}\rvert}=4\left(\frac{\sqrt{14}}{14}\right)\begin{pmatrix}1\\2\\3\end{pmatrix}=\frac{2\sqrt{14}}7\begin{pmatrix}1\\2\\3\end{pmatrix}

A positive multiplier keeps the direction. Multiply the unit vector, rather than the original position vector, to obtain the requested length.

How did this part go?

Common mistakes

Multiplying the original vector by 4 would give length 4√14, not 4.