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Main Question or Discussion Point
Dear all,
My question is from the text of Alan F. Beardon, Algebra and Geometry concerning the scalar triple product. I have attached the text in this post.
In order for the STP to be nonzero. The 3 vectors must be distinct and they are not coplanar. 2 vectors can be coplanar and only the 3rd vector has to lie on a different plane right??
Geometrically, STP can be interpreted as the volume of the parallelepiped formed by the 3 vectors.
Now, STP is also used to define a point x with reference to any given set of nonorthogonal coordinates axes along the directions of a, b & c. a, b & c can be considered to be like unit vectors right? Can two of the line segments [0,a], [0,b] or [0,c] lie on the same plane??
The formula in 4.4.3 shows that point x can be written in terms of the 3 axes vectors, a,b & c. The scalar multiple of each vector is
## \dfrac{[x,b,c]}{[a,b,c]} ##
I am trying to understand how this came about!! all 3 a,b & c have a common denominator ## [a,b,c] ##. b cross with c is orthogonal to b and c and it is dot with a. This is like the component of a in the direction of b cross c.
because for any 2 vectors L and M
## L \cdot M = L M Cos θ ##
Is there a better way to understand/interpret this as it is used for other two components b and c as well?
Next, the numerator for a, b and c are as follows ## [x,b,c], [a,x,c], [a,b,x] ##
## [x,b,c] ## can be understood as the component of b in the direction b cross c. ## [a,x,c] & [a,b,x] ## can be understood likewise. But I don't get the general idea as to why we are doing this?? I cannot use the geometrical interpretation of the volume of a parallelepiped.
Also please show how the following is derived so I can apply this to get μ and ν.
## [x,b,c] = [λa + μb + νc, b, c] = λ[a,b,c] ##
Danke....
My question is from the text of Alan F. Beardon, Algebra and Geometry concerning the scalar triple product. I have attached the text in this post.
In order for the STP to be nonzero. The 3 vectors must be distinct and they are not coplanar. 2 vectors can be coplanar and only the 3rd vector has to lie on a different plane right??
Geometrically, STP can be interpreted as the volume of the parallelepiped formed by the 3 vectors.
Now, STP is also used to define a point x with reference to any given set of nonorthogonal coordinates axes along the directions of a, b & c. a, b & c can be considered to be like unit vectors right? Can two of the line segments [0,a], [0,b] or [0,c] lie on the same plane??
The formula in 4.4.3 shows that point x can be written in terms of the 3 axes vectors, a,b & c. The scalar multiple of each vector is
## \dfrac{[x,b,c]}{[a,b,c]} ##
I am trying to understand how this came about!! all 3 a,b & c have a common denominator ## [a,b,c] ##. b cross with c is orthogonal to b and c and it is dot with a. This is like the component of a in the direction of b cross c.
because for any 2 vectors L and M
## L \cdot M = L M Cos θ ##
Is there a better way to understand/interpret this as it is used for other two components b and c as well?
Next, the numerator for a, b and c are as follows ## [x,b,c], [a,x,c], [a,b,x] ##
## [x,b,c] ## can be understood as the component of b in the direction b cross c. ## [a,x,c] & [a,b,x] ## can be understood likewise. But I don't get the general idea as to why we are doing this?? I cannot use the geometrical interpretation of the volume of a parallelepiped.
Also please show how the following is derived so I can apply this to get μ and ν.
## [x,b,c] = [λa + μb + νc, b, c] = λ[a,b,c] ##
Danke....
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