WO2023096507A1

WO2023096507A1 - 3-d instructional apparatus for vector operations

Info

Publication number: WO2023096507A1
Application number: PCT/PH2022/050024
Authority: WO
Inventors: Cecilia BUCAYONG
Original assignee: Central Mindanao University
Priority date: 2021-11-25
Filing date: 2022-11-23
Publication date: 2023-06-01

Abstract

A 3-D instructional apparatus that could demonstrate vectors in 3-D space showing its projections along x, y, and z axes. Calibrations on posts and movable sticks anchored in grooves can represent the magnitude and component of vectors. The apparatus's eight octants coordinate system embodiment would enable vector operations in 3-D experimentally. Thus, this device aids in visualizing vectors in Three Dimensions and can perform highly mathematical vector operations experimentally with comparable results using the analytical method.

Description

Title of the Invention: 3-D Instructional Apparatus for Vector Operations

Technical Field

The present invention pertains to a three-dimensional (3-D) instructional apparatus, particularly an apparatus for experimentally visualizing and performing vector operations.

Background Art

Vectors given in three dimensions are difficult to visualize using two-dimensional chalkboards or writing pads. The problem is compounded with added analytical computations as required in this topic, and it would become much harder if the instructional approach were purely theoretical. Thus, physics teaching should include demonstrations and experiments.

A typical instructional device used to illustrate and perform vector operations is a "force table." However, the "force table" described in the prior art (US1806350A) is limited to two-dimensional vectors only. Further, the said device is limited to vector addition and cannot be used for vector products.

Another prior art is the "Apparatus for simulating 3-D dynamic deforming vector" (CN1862285A), a 3-D apparatus. However, this is only limited to simulating 3-D vector deformation and trajectories, not for vector operations. The same reasoning applied to the "Three-dimensional vector co-processor having I, J, and K register files and I, J, and K execution units" (US5187796A) invention. The invention is a three-dimensional vector co-processing system (3DVCP) used individually for scalar operations, not for the instructional purpose of adding and multiplying vectors, as claimed in the present invention.

It is, therefore, an objective of the present invention to create a 3-D instructional device that could help improve the teaching and learning endeavors of teachers and students, respectively. Furthermore, this 3-D apparatus could provide visuals of vectors regardless of their position in space. The students, therefore, will be ushered towards conceptual understanding amidst highly computational subject matter like vector products. Summary of Invention

With its mechanical features, the present invention helps to meet certain drawbacks of the existing instructional device used in illustrating and solving vector operations. For example, this 3-D instructional apparatus could illustrate vectors given in three dimensions and can be used in solving vector addition, dot, and cross products. Moreover, this invented apparatus was designed to form a rectangular coordinate system in eight octants representing a vector in 3-D space. In this manner, vectors can be visualized and positioned anywhere in space as long as they are within the said coordinate system.

Brief Description and Drawings

A better understanding of the present invention is hereto referred to in the following brief descriptions and accompanying drawings.

Fig.1

[fig.1 ] is the schematic illustration of the 3-D Apparatus, according to some embodiments of the present invention.

Fig.2

[fig .2] illustrates the top view of Figure 1 .

Fig.3

[fig.3] is the side view of Figure 1 .

Fig.4

[Fig.4.] is a bottom sectional view of Figure 1 , showing the grooves of the corner vertical column and the opposing grooves of the middle vertical column.

Fig.5

[fig.5] is the schematic illustration of the 2^nd groove provided on each opposing side of the central vertical support disposed on each side of the cubical frame; Detailed Description of the Invention

A. Construction of the 3-D Instructional Apparatus for Vector Operations

The needed materials for constructing the 3-D Instructional Apparatus for Vector Operations consist of the following. i. 4 square bars (20) grooved only at two connecting sides ii. 4 square bars (50) grooved at opposing sides iii. 8 non-grooved square bars (30) iv. 2 square bars (70,80) with a hole at mid-length v. 1 central vertical shaft (100) - dimension fits the hole of the said two square bars (70,80) vi. Plurality of shorter square bars (90) with endpoints fitting to the grooves vii. Plurality of strings (110) viii. Plurality of detachable strings (111)

First is to assemble a cubical frame with four square bars (20) at the vertical corners and four square bars (50) disposed of centrally and vertically in between. All the said vertical bars are connected to non-grooved square bars (30) connected horizontally at the top and bottom sides of the skeletal frame (10). The remaining two square bars (70, 80) with a hole are then connected horizontally at the top and bottom of the skeletal frame. The central vertical shaft (100) is positioned as the central vertical post inserted at the holes of square bars (70, 80) on the top and bottom sides of the skeletal frame (10). A plurality of strings (110) is securely connected at the mid-length (120) of the central shaft (100). Except for the non-grooved square bars (30), all the remaining square bars; the vertical and horizontal, and the shorter bars are provided with markings for predetermined measurement. Shorter square bars (90) are only connected as needed in the actual use of the apparatus. The strings (110) will be anchored to the said shorter square bars (90) using clips to represent vectors.

B. Composition and Function

Figure 1 shows the invention's preferred embodiment composed of a skeletal frame (10), which defines the rectangular coordinate system with eight octants. The central vertical shaft (100) represents the y-axis, with its mid-length marked as the origin. In addition, the vertical supports at the corners (20) and central supports (50) on the sides of the rectangular frames also define the y- components. The x-axis components are defined by the top (70) horizontal supports and all the adjustable horizontal bars (90) along with the said supports. The bottom horizontal support (80) and all the adjustable horizontal bars (90) along the said support define the z-axis components. The magnitude of a given vector to be added or multiplied and the resultant is determined by directly measuring the string (1 10) representing the said vector. The components of vectors are determined by simply reading the markings provided on the vertical corner supports (20), central vertical support (50), a central vertical shaft (100), and adjustable horizontal bars (90). The markings with predetermined measurements represent the number line along different axes (x, y, z) of a cartesian coordinate system illustrated by the apparatus.

C. Vector Addition using the Apparatus

Strings (110) already attached from the origin (120) represent vectors to be added. A sample given vector A can be positioned accurately following its dimensions and direction since the adjustable horizontal bars (90) were designed to slidably disposed between the first vertical grooves (40) and the second vertical grooves (60). A marker(anything) should be attached to the string (1 10) to indicate the vector head. To position the vector according to its given dimension, strings (11 1) - not the plurality of strings attached from the origin, and clips will be used. The string (1 11) should always be anchored perpendicularly across horizontal bars (90) to correctly position and fix the vector head on the said string (1 10). The magnitude of the vector is the length of the string (110), measured from the origin (120) to its vector head (marker). The said length is equal to the mathematical expression /A/ = Vn² + b². Using the apparatus, the scaling in measuring the magnitude of the vector must be consistent with the markings on the bars. From the head of the vector, the lengths of its projection along the x, y, and z axes correspond mathematically to its A_xi, A_yj, and A_zk components, respectively. Using the apparatus, the A_xi is the length along the apparatus designated x-axis measured from the origin to the perpendicular string which anchored the vector's head. The length represents the x-component or the projection of the given vector along the x-axis. In the same manner, A_yj and A_zk or the component of the vector along the y and z axes, respectively, are the lengths along its corresponding axes, measured from the origin to the string attached perpendicularly fixing the vector's head. Thus, the analytical expression of a sample vector A in 3-D denoted as A = A_xi+ A_yj + A_zk can be illustrated and directly measured using the 3-D apparatus. The direction of vector A mathematically expressed as a_A =

y_A = — cos^-1. Using the 3-D apparatus, the said angles are measured by a protractor without computations. The a_A is equivalent to the angle between the string representing vector A and the horizontal bars along the designated x-axis. In the same manner, _A and y_A are the angles between the string representing vector A and the bars along the designated y-axis and z-axis, respectively. Thus, the apparatus can visualize the 3D position of a vector and can directly measure the angles a_A, f>_A and y_A using a protractor.

Other given vectors, denoted as vectors B, C, and so on, follow the same positioning and measurements according to their correct dimension and position. In cases where the magnitude exceeds the markings of the apparatus, appropriate scaling must be followed.

In using performing vector addition using the apparatus, the resultant magnitude of two vectors is determined with the help of strings (11 1) representing A prime (A') and B prime (B'). A' is equivalent to vector A (same length and direction as vector A) but connected from the head of vector B. Likewise, B' is the equal vector B (same length and direction as vector B) but connected from the head of vector A. Thus A' is a string parallel to vector A, and B' is a string parallel to vector B. Following their correct magnitude and direction, the two primed vectors are expected to intersect at a common point. The resultant vector is represented by a string (1 10) from the origin (120) connected to the point of intersection of the primed vector's A' and B'. The magnitude of the resultant is equal to the length of the said string or the length of the line connecting the origin (120) and the intersection point of these primed vectors A' and B'. Mathematically, this is equivalent to /A + B/= (Ax + Bx)² + (Ay + By)² . Components of the resultant vector (R_x, R_y, R_z) are the lengths of its projection along x, y, and z axes, respectively. Using the apparatus, these correspond to the lengths along x, y, z designated axes, measured from the origin (120) up to the strings attached perpendicularly to fix the intersection point of primed vectors. The direction of the resultant is determined using a protractor. The measured angle between x-axis and the string representing the resultant vector is the a_R. Similarly, the measured angle between y-axis and the resultant is p_B, and the measured angle between z-axis and the resultant is y_R. The mathematical equivalents are as follows; a_R = — R cos^-1, p_B =

□.Vector Multiplication Using the Apparatus

1 . Dot Product

The dot product equation is equal to A. B = A_XB_X + A_yB_y + A_ZB_Z or A. B = ABcos Q , the angle between these two vectors being multiplied is © = — ^A

'^B Using the apparatus, connect a string (111) to the two strings (110) representing the two vectors A and B. The said string (11 1) must originate from the head of vector A and passing perpendicularly to vector B, or vice versa, because dot product is commutative. The three strings now form a right triangle. The dot product magnitude is equal to the length measured from the origin to the base of the said right triangle, times the magnitude of the vector along with it (either A or B). If along A vector, such length is equal to B cos ©, otherwise is A cos ©. This length is the component of vector B along A or the component of A along B. The lengths can be measured directly using an appropriate measuring device, and the angle © is measured using a protractor. . Cross Product

The analytical equation of finding the magnitude of the cross product Ax B = ABsin 0. This mathematical equation can be visualized using the apparatus. Performing the steps in adding two vectors A+B by connecting strings as primed vectors A' and B.' The four strings representing A, B, A' and B' form a parallelogram. The area of this parallelogram is equal to the magnitude of the cross product A x B. To determine the area (base times height), measure the line length originating perpendicularly from the first vector A and ending on the head of the second vector B. The measured length is the height or B sin ©. The base of the parallelogram is the length of the string representing vector A. Note that this is not the same base (B cos ©) in the dot product, but this is the altitude of the said triangle. Cross-product is not commutative because it affects the direction of the product. Thus, A x B is not equal to B x A. Note that in cases where the product magnitude exceeds the calibration of the apparatus, appropriate scaling is followed.

In positioning the string representing the cross-product direction, an L-square instrument is suggested. The line representing the product is positioned so that it appears to originate perpendicularly from the area of the plane formed by the vectors being multiplied. Since the said plane has two faces, the specific direction of this product is determined using the Right-Hand Rule.

The components of the vector product are obtained experimentally by directly measuring the projections of this vector along the x, y, and z axes. The measured length of the projections equals the magnitude of their respective components.

Claims

CLAIMS An instructional apparatus for vector operations comprising: a skeletal frame (10), said skeletal frame (10) being defined by four vertical corner supports (20), each of said vertical corner supports (20) being provided with first vertical grooves (40) disposed at two inner sides of said vertical corner support (20), non-grooved horizontal support (30) disposed at the top and bottom sides of the skeletal frame (10); a second vertical support (50) disposed centrally between the first vertical supports (20) on each side of said skeletal frame (10), said central vertical support (50) being provided with second vertical grooves (60) on each opposing side; a horizontal support (70, 80) disposed centrally at the top and bottom side of said skeletal frame (10) connecting a pair of opposing vertical support (50) at least an adjustable horizontal bar (90) slidably disposed between said first vertical groove (40) and said second vertical groove (60); a central vertical shaft (100) disposed centrally inside said skeletal frame (10) connecting said top horizontal support (70) and said bottom horizontal support (80) at midpoints; and a plurality of strings (110) suspended from said central vertical shaft (100) at mid-length (120) thereof. The instructional apparatus for vector operations according to claim 1 , said vertical corner supports (20), said central vertical support (50), said central vertical shaft (100), and said adjustable horizontal bars (90) are being provided with markings pertaining to predetermined measurement.