• This lesson covers lessons 3-1 and 3-2 of the unit "Geometry and Measurement in Two and Three Dimensions".
• After it you should be able to:
• locate a point in space and find the distance between two points and the midpoint of a segment;
• represent a vector by three components and carry out addition, scalar multiplication and equality in space;
• find the norm, the unit vector, the direction angles and direction cosines of a vector.

🌳 Concept tree

The 3D Coordinate System and Vectors in Space

• ├── 3-1 (a) The three-dimensional orthogonal coordinate system

│ ├── Why three axes

│ ├── The right-hand rule

│ ├── The coordinate planes

│ ├── Plotting a point

│ └── Distances to planes and axes

├── 3-1 (b) Distance between two points

│ └── Rule

├── 3-1 (c) The midpoint of a segment

│ └── Rule

├── 3-2 (a) Position vectors and components

│ └── Definitions

├── 3-2 (b) Operations on vectors

│ ├── Addition

│ ├── Multiplication by a real number

│ └── Equality of vectors

• ├── 3-2 (c) Unit vectors and the basis $\widehat{i}$, $\widehat{j}$, $\widehat{k}$

│ ├── The fundamental unit vectors

│ └── Unit vector

├── 3-2 (d) The vector between two points

│ └── Rule

• └── 3-2 (e) Direction angles and direction cosines

├── Definitions

└── Useful facts

Definitions

• The position vector of $A(A_x,A_y,A_z)$ is the directed segment $\overrightarrow{OA}$ from the origin to $A$, written $\vec{A}=(A_x,A_y,A_z)$.
• $A_x$, $A_y$, $A_z$ are the components of the vector along the three axes.
• The norm (magnitude) of A⃗ is its length ∥A⃗∥=Ax2+Ay2+Az2
• Example for A⃗=(2,3,1) the norm is 14 and for B⃗=(0,4,−3) it is 5.
• A vector with one zero component lies in a coordinate plane; for example $\vec{B}$ above lies in the $yz$-plane.

• Let ${\theta }_x$, ${\theta }_y$, ${\theta }_z$ be the angles a vector $\vec{A}$ makes with the positive $x$, $y$, $z$ axes.
• They are the direction angles; $\cos {\theta }_x$, $\cos {\theta }_y$, $\cos {\theta }_z$ are the direction cosines, often written $l$, $m$, $n$.
• The components are $A_x=\mathrm{\parallel }\vec{A}\mathrm{\parallel }\cos {\theta }_x$, $A_y=\mathrm{\parallel }\vec{A}\mathrm{\parallel }\cos {\theta }_y$, $A_z=\mathrm{\parallel }\vec{A}\mathrm{\parallel }\cos {\theta }_z$.
• So the unit vector is ${\widehat{u}}_A=(\cos {\theta }_x,\cos {\theta }_y,\cos {\theta }_z)$, and therefore
  $${\cos }^2{\theta }_x+{\cos }^2{\theta }_y+{\cos }^2{\theta }_z=1$$
• Each direction angle lies between $0^{\circ }$ and $180^{\circ }$.

✍️ Worked examples

• Example 15 Find the direction cosines and angles of $\vec{A}=(1,2,2)$.
• Solution:
• $\mathrm{\parallel }\vec{A}\mathrm{\parallel }=3$.
• Direction cosines 13, 23, 23 (their squares add to 1+4+49=1).
• Angles cos⁡−113≈70.5∘, and cos⁡−123≈48.2∘ twice.
• Example 16 A force of 10 N makes angles of $60^{\circ }$ with $Ox$ and $45^{\circ }$ with $Oy$ and an acute angle with $Oz$. Write it in component form.
• Solution:
• ${\cos }^{2}60^{\circ }+{\cos }^{2}45^{\circ }+{\cos }^2{\theta }_z=1$ gives ${\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}+{\cos }^2{\theta }_z=1$, so $\cos {\theta }_z={\displaystyle \frac{1}{2}}$ and ${\theta }_z=60^{\circ }$.
• $\vec{F}=10({\displaystyle \frac{1}{2}},{\displaystyle \frac{\sqrt{2}}{2}},{\displaystyle \frac{1}{2}})=(5,5\sqrt{2},5)\approx (5,7.07,5)$ N.
• Example 17 A vector of norm 5 makes equal angles with the three positive axes. Write it in component form.
• Solution:
• Equal angles give $3{\cos }^2\theta =1$, so $\cos \theta ={\displaystyle \frac{1}{\sqrt{3}}}$ and $\theta \approx 54.7^{\circ }$.
• $\vec{A}=5({\displaystyle \frac{1}{\sqrt{3}}},{\displaystyle \frac{1}{\sqrt{3}}},{\displaystyle \frac{1}{\sqrt{3}}})={\displaystyle \frac{5}{\sqrt{3}}}(1,1,1)\approx (2.89,2.89,2.89)$.
• Example 18 Resolve a vector of norm 10 making $40^{\circ }$ with $Oz$, whose projection on the $xy$-plane makes $70^{\circ }$ with $Ox$.
• Solution:
• $A_z=10\cos 40^{\circ }\approx 7.66$ and the projection on the $xy$-plane has length $10\sin 40^{\circ }\approx 6.43$.
• $A_x=6.43\cos 70^{\circ }\approx 2.20$ and $A_y=6.43\sin 70^{\circ }\approx 6.04$.
• $\vec{A}\approx 2.20\widehat{i}+6.04\widehat{j}+7.66\widehat{k}$.
• Direction angles cos⁡−12.2010≈77.3∘, cos⁡−16.0410≈52.8∘, 40∘.