### About

#
*3D* Visualization of vector calculus

This simulation visualizes the basic vector operations in* 3D*.
Activating the button * new vectors* generates vectors of
random orientation, drawn as blue arrows. For better clarity all arrows
have an absolute value (length) of 1 and start at the magenta colored
origin. They are embedded into a sphere of radius 1, that is shown as a
transparent mesh.

At the start two vectors will be visible, *a* with red filling
color of the arrowhead, *b *with white color.

At the left side one can chose among different perspective projections
by **radio buttons:**

**perspective: ***3D *projection with perspective distortion. It
can be rotated with the mouse.

**xy****-projection**: top view of the *xy*-plane
(along the *z-*axis)

**yz****-projection**: along the *x*-axis

**xz****-projection**: along the *y*-axis

**no perspective**: *3D* projection without distortion. It can
be rotated with the mouse.

At the bottom the following data are shown for the 2 vectors *a *and
*b *

**Intersecting angle:** in degrees.

**Product of absolute values:** always 1, as both are of length 1.

**Scalar product:** **a∙b** = **a| |b| **cos(**a|b**),
with **a|b** the intersecting angle.

**Absolute value of the vector product:** |**a x b**| = **|a| |b| **sin(**a|b**).

At the top the following operations can be activated by **check boxes:**

**Addition:** **a + b.**

**Subtraction: a **-** b.**

**Subtraction: b **-** a (= **-**(a **-** b)).**

**Vector product: a **x** b.**

**Vector product: b **x** a (= **-** a **x** b).**

**3 Vectors a + b + c: **generation of *c* and sum (**c with
yellow filling color**).

The chosen combination is maintained when new vectors are generated.

# Vector calculus

Vectors

*a = (a _{1 }, a_{2 }, a_{3}) *

*b = (b _{1 }, b_{2 }, b_{3})*

**Absolute value** (length of the vector arrow) *|a| = √(a _{1}^{2}
+ a_{2}^{2 }+ a_{32 })*

**Addition** **a **+** b** *= (a _{1}+b_{1 },a_{2}+b_{2
}, a_{3}+b_{3}) =*

**b**+

**a**

**Subtraction a **-** b** = *(a _{1 }-b_{1}
,a_{2}- b_{2 }, a_{3}- b_{3})= -*

**(b**-

**a)**

**Subtraction b **-** a **= *(b _{1}- a_{1 }, b_{2}-
a_{2 }, b_{3}- a_{3})* =

*-*

**(a**-

**b)**

**Multiplication by a constant ***k: k***a** = *(ka _{1 },
ka_{2 }, ka_{3})*

**Skalar product (internal product)** **a · b** = *a _{1}b_{1}
+a_{2}b_{2} + a_{3}b_{3} =*

**|a| |b|**cos(

**a|b**)

**a** perpendicular to **b** ⇒ **a|b ***= 90 ^{o}
*⇒cos(

**a|b**)

*= 0 ⇒ Scalar Product = 0*

**Vector product (external product) a **x** b** = *(a _{2}b_{3}-b_{2}a_{3
}, a_{3}b_{1} - b_{3}a_{1 }, a_{1}b_{2}
- a_{2}b_{1})*

**Vector product b **x** a**= *(b _{2}a_{3}- a_{2}b_{3}
, b_{3}a_{1} - a_{3}b_{1 }, b_{1}a_{2}
- a_{1}b_{2})* = -

**a**x

**b**

**Absolute value of vector product** |**a **x** b|** = |**a**|
|**b**| sin(**a|b**)

**a **parallel **b **⇒ **a|b **= 0^{o} ⇒sin(**a|b**)
= 0 ⇒ |**vp**| = 0

**a **x** b and b **x** a **are perpendicular** **to the
plane common to **a** and **b**

**a , b **and **a **x** b **form a clockwise tripod

**a, b, **and **b **x** a** form a counter clockwise tripod

when going from **a** to **b **and then to the **vector product.**

**E1:** Generate random vectors. Rotate the sphere to have a top down
look at their common plane. Estimate the angle between the vectors and
compare your estimate to the value displayed in the bottom line. **Experiment
with the different views **and reflect well what you see with every
one of them**. **

**E2:** The product of absolute values is displayed to be 1 (each of
the 2 vectors has an absolute value of 1). Compare that to the scalar
product. When is the scalar product about the same as the absolute one?
When is it zero? Observe the display of the intersecting angle.

**E3:** Select **a + b** in the upper row. A thin red vector is
affixed to * a*, whose length and direction are those of

*. A thick red vector starting at the origin is the sum vector. Describe its geometric construction. (mentally complete the 3 vector to a parallelogram ). Convince yourself by rotation of the frame, that*

**b**

**a,***and*

**b***are in one plane.*

**a+b**

**E4: **Rotate the frame and** **observe that the parallelogram
construction is valid for every projection.

**E5:** As E3, but add **a + b + c** . You will see a third
vector * c* und a sum vector

*, besides the vector*

**a+b+c****a+b.**Observe that

**a, b**und

**c**in general will not lie in a plane, but form a tripod with non perpendicular axes. Perform in mind the geometric construction from

**a+b**to

**a+b+c**.

**E5: **Close other options and choose** a **-** b**. A thin
complement vector will appear in magenta. Convince yourself by rotating
that this construction also works for all projections. Remember that **a
**-** (**-**b) = a + b**.

**E6:** Chose **b **- **a **and** **interpret what **b **-**
a = **-** (a **-** b)** means geometrically.

**E7:** Close other options and choose the vector product** a** x**
b **which will appear as a black arrow. Rotate the frame to look along
the vector product. Observe that you will look perpendicular at the
common plane of **a **and** b** (both vectors should touch the
sphere periphery).

**E8: **Choose *new vectors* several times and note when the
absolute value of the vector product is very small or very large. What
is the criterion? What are the possible minimum and maximum values? (**a
**and **b **both have the absolut value of 1).

**E9: **Chose** b **x** a** and interpret what **b **x** a
= **-** a **x** b **means geometrically.

**E10:** When rotating the frame the length of the vectors and their
relativ orientation stay unchanged. *They are invariant under
rotation. *They are also invariant under shift. Confirm this mentally
(remember the compensating vectors of the construction)

This file was created in January 2009 by Dieter Roess

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH- Foundation; Fremont Teng; Loo Kang Wee

### end faq

### Sample Learning Goals

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### For Teachers

## Vectors Operations in 3D JavaScript Simulation Applet HTML5

### Instructions

#### New Vectors

#### Combo Box and their Functions

#### Toggling Full Screen

#### Reset Button

Research

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### Video

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### Version:

### Other Resources

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