Does anyone know where to download a programme that will allow me to visualise the 3-dimensional/geometrical representation of the vectors (ie not just the simple 3-D vectors but also include the 3D of cross product, components of the vectors, vectors as in plane(s) and so on) as in H2 Maths ?
Thanks.
Originally posted by Snoopyies:Does anyone know where to download a programme that will allow me to visualise the 3-dimensional/geometrical representation of the vectors (ie not just the simple 3-D vectors but also include the 3D of cross product, components of the vectors, vectors as in plane(s) and so on) as in H2 Maths ?
Thanks.
Hi,
Why don't you draw them out on paper to enable visualisation?
The topic of vectors in H2 Maths can be quickly summarised into these 8 points:
1. Collinearity of 3 points and Ratio Theorem.
2. Definition of the scalar product,
i.e. a . b = |a| |b| cos \theta.
This definition is useful for finding length of projection of one vector onto another, angle between 2 vectors, angle between 2 lines, angle between 2 planes and angle between line and plane.
3. Definition of the magnitude of the vector product,
i.e. |a x b| = |a| |b| sin \theta.
This definition is useful for finding shortest distance between point and line as well as areas of figures.
4. The normal c is parallel to a x b means that c is perpendicular to each of a and of b. This idea is extremely helpful for finding equation of a plane containing a line and is parallel to some vector. It is also useful to understand the line of intersection between two planes.
5. The relationship between two lines may be intersecting, parallel or skew.
6. The relationship between three planes may be intersecting at a point, intersecting at a line, parallel or non-intersecting.
7. The relationship between a line and a plane may be intersecting, parallel or containment (i.e. line is on the plane).
8. Equation of a line may be expressed in vector, parametric or cartesian forms. Equation of a plane may be expressed in vector, scalar product or cartesian forms.
Thanks!
Cheers,
Wen Shih
Originally posted by Snoopyies:Does anyone know where to download a programme that will allow me to visualise the 3-dimensional/geometrical representation of the vectors (ie not just the simple 3-D vectors but also include the 3D of cross product, components of the vectors, vectors as in plane(s) and so on) as in H2 Maths ?
Thanks.
http://software.filestube.com/software,f61e00ef,OxMath+Classes.html
Originally posted by MasterMoogle:
Hi MasterMoogle,
Thanks for the url link.
Originally posted by wee_ws:Hi,
Why don't you draw them out on paper to enable visualisation?
The topic of vectors in H2 Maths can be quickly summarised into these 8 points:
1. Collinearity of 3 points and Ratio Theorem.
2. Definition of the scalar product,
i.e. a . b = |a| |b| cos \theta.
This definition is useful for finding length of projection of one vector onto another, angle between 2 vectors, angle between 2 lines, angle between 2 planes and angle between line and plane.3. Definition of the magnitude of the vector product,
i.e. |a x b| = |a| |b| sin \theta.
This definition is useful for finding shortest distance between point and line as well as areas of figures.4. The normal c is parallel to a x b means that c is perpendicular to each of a and of b. This idea is extremely helpful for finding equation of a plane containing a line and is parallel to some vector. It is also useful to understand the line of intersection between two planes.
5. The relationship between two lines may be intersecting, parallel or skew.
6. The relationship between three planes may be intersecting at a point, intersecting at a line, parallel or non-intersecting.
7. The relationship between a line and a plane may be intersecting, parallel or containment (i.e. line is on the plane).
8. Equation of a line may be expressed in vector, parametric or cartesian forms. Equation of a plane may be expressed in vector, scalar product or cartesian forms.
Thanks!
Cheers,
Wen Shih
Hi Wee_ws,
Thanks for the very useful summary on the topic Vector.
However, I have difficulties to understand the topic Vector.
1. What does Dot/Scalar Product a.b = lal.lbl.cos theta mean (textbook just gives us the formula and the diagram and it does not explain why a.b will be equal to lal.lbl.cos theta and it also does not explain the meaning of a.b or lal.lbl.cos theta ?
2. What does cross product of vectors la x bl = lal.lbl.sin theta mean (the textbook does not explain why la xbl will be equal to lal.lbl.sin theta and it also does not explain why the multiplication of the two vectors a and b on the horizontal plane will lead to the third vector on the vertical plane ? The textbook also did not explain a normal vector, it just says that for a plane parallel to vectors b and vector c, the vector b x c will be the normal vector to the plane ie why the normal vector is b x c.
3. Based on the diagram in the textbook, it is difficult to visualise the 3 components of a vector, it will be good if there is a software that will allow us to visual it in 3D with motion function ie it can be rotated. It will be so much easier to understand the H2 maths vector topic especially on those parts involving planes and lines.
Thanks for the very useful summary and help.
Originally posted by Snoopyies:Hi Wee_ws,
Thanks for the very useful summary on the topic Vector.
However, I have difficulties to understand the topic Vector.
1. What does Dot/Scalar Product a.b = lal.lbl.cos theta mean (textbook just gives us the formula and the diagram and it does not explain why a.b will be equal to lal.lbl.cos theta and it also does not explain the meaning of a.b or lal.lbl.cos theta ?
2. What does cross product of vectors la x bl = lal.lbl.sin theta mean (the textbook does not explain why la xbl will be equal to lal.lbl.sin theta and it also does not explain why the multiplication of the two vectors a and b on the horizontal plane will lead to the third vector on the vertical plane ? The textbook also did not explain a normal vector, it just says that for a plane parallel to vectors b and vector c, the vector b x c will be the normal vector to the plane ie why the normal vector is b x c.
3. Based on the diagram in the textbook, it is difficult to visualise the 3 components of a vector, it will be good if there is a software that will allow us to visual it in 3D with motion function ie it can be rotated. It will be so much easier to understand the H2 maths vector topic especially on those parts involving planes and lines.
Thanks for the very useful summary and help.
Very inquisitive of you. But what you ask is what my shi fu is learning in his uni 1st year now. Maybe next time you will learn them too.
Originally posted by Snoopyies:Hi Wee_ws,
Thanks for the very useful summary on the topic Vector.
However, I have difficulties to understand the topic Vector.
1. What does Dot/Scalar Product a.b = lal.lbl.cos theta mean (textbook just gives us the formula and the diagram and it does not explain why a.b will be equal to lal.lbl.cos theta and it also does not explain the meaning of a.b or lal.lbl.cos theta ?
2. What does cross product of vectors la x bl = lal.lbl.sin theta mean (the textbook does not explain why la xbl will be equal to lal.lbl.sin theta and it also does not explain why the multiplication of the two vectors a and b on the horizontal plane will lead to the third vector on the vertical plane ? The textbook also did not explain a normal vector, it just says that for a plane parallel to vectors b and vector c, the vector b x c will be the normal vector to the plane ie why the normal vector is b x c.
3. Based on the diagram in the textbook, it is difficult to visualise the 3 components of a vector, it will be good if there is a software that will allow us to visual it in 3D with motion function ie it can be rotated. It will be so much easier to understand the H2 maths vector topic especially on those parts involving planes and lines.
Thanks for the very useful summary and help.
Hi,
At your level of study, you can only accept a . b and |a x b| as definitions.
We can use real-life objects to visualise intersection of 3 planes at a point, say a cube whose three surfaces meet at a vertex, or two surfaces meet along an edge.
Thanks!
Cheers,
Wen Shih
Dot product of two vectors is the magnitude of one vector times the projection of the other vector along the first vector ie
a.b = lal x the projection of the vector b on the direction of the vector a
Using CAH, the projection of the vector b on the direction of the vector a is lbl.cos theta.
Hence, a.b = lal.lbl.cos theta
Magnitude of the Cross product of two vectors is the area of the parallelogram between them ie
la x bl = area of the parallelogram
la x bl = base x height
la x bl = lal x lbl sin theta (height is found by using SOH)
Hi Lee012lee,
Thanks, trigo and geometrical concepts can enhance a student's understanding of dot and vector products :)
To Snoopyies:
1. Draw a right-angled triangle.
2. Let the hypothenuse be represented by vector a.
3. Let the adjacent side by represented by vector b.
4. Let the angle between vectors be \theta.
5. Length of projection of a onto b is given by
|a| cos \theta,
the horizontal component.
This simplifies to |a . b| / |b|, by definition of scalar product.
Note that |a . b| refers to the absolute value of the scalar product, since it may be a negative value and length cannot be negative.
6. Shortest distance of point A from the line (going in the direction of b) is given by
|a| sin \theta,
the vertical component.
This simplifies to |a x b| / |b|, by definition of magnitude of vector product.
Note that |a x b| refers to the magnitude of the vector a x b.
Cheers,
Wen Shih