Algebraic vector problems (planes, cross products etc)

[PedroB]

Homework Statement

1. A vector u is given by u = λa + μb, where λ, μ are elements of ℝ. Calculate axu and bxu and show that:

$[(b χ u).n]/[(b χ a).n]$ = λ

and

$[(a χ u).n]/[(a χ b).n]$ = μ

(In any circumstance when I have omitted the underlining it is for convenience purposes, a, b, n and u are still vectors.)


2. The vectors a and b form a set of basis vectors for P₁ (A plane spanned by vectors a and b). However it is often desirable to use a set of orthogonal basis vectors.

Construct a vector b' that is in P ₁ and orthogonal to a.

Homework Equations

We are told that a and b are vectors in ℝ³, and that they are not parallel or anti-parallel. n is the normal vector to plane P₁.

The Attempt at a Solution

1. n= a x b

a x b = a x (aλ x bμ)
= a x bμ
= nμ

∴$(a χ u)/n$ = μ
$(a χ u)/(a χ b)$ = μ

The same can be applied for obtaining an equation with λ. The problem arises out of the fact that I wouldn't know how to incorporate the '.n' portion to the numerator and denominator (though it seems quite obvious that their cancellation would yield the above equation, it must serve some purpose).

2. I can't find any sort of methodology that would yield a vector, b', perpendicular to a in P₁ when only given the information above. I would assume it has something to do with question 1 (I could be wrong), though I can't see how they relate to one another.


Thanks in advance for any help.

 

[Mark44]

Mod note: I tweaked what you wrote to make it more readable. In particular, I replaced the character you used for the cross product (Greek letter chi?) with X.

Homework Statement

1. A vector u is given by u = λa + μb, where λ, μ are elements of ℝ. Calculate axu and bxu and show that:

[(b X u) $\cdot$ n]/[(b X a) $\cdot$ n] = λ

and

[(a X u) $\cdot$ n]/[(a X b) $\cdot$ n] = μ

What is n?

(In any circumstance when I have omitted the underlining it is for convenience purposes, a, b, n and u are still vectors.)2. The vectors a and b form a set of basis vectors for P₁ (A plane spanned by vectors a and b). However it is often desirable to use a set of orthogonal basis vectors.

Construct a vector b' that is in P ₁ and orthogonal to a.

Homework Equations

We are told that a and b are vectors in ℝ³, and that they are not parallel or anti-parallel. n is the normal vector to plane P₁.

The Attempt at a Solution

1. n= a x b

a x b = a x (aλ x bμ)
= a x bμ
= nμ

∴$(a X u)/n$ = μ
$(a X u)/(a X b)$ = μ

The same can be applied for obtaining an equation with λ. The problem arises out of the fact that I wouldn't know how to incorporate the '.n' portion to the numerator and denominator (though it seems quite obvious that their cancellation would yield the above equation, it must serve some purpose).

2. I can't find any sort of methodology that would yield a vector, b', perpendicular to a in P₁ when only given the information above. I would assume it has something to do with question 1 (I could be wrong), though I can't see how they relate to one another.Thanks in advance for any help.

 

[LCKurtz]

Homework Statement

1. A vector u is given by u = λa + μb, where λ, μ are elements of ℝ. Calculate axu and bxu and show that:

$[(b χ u).n]/[(b χ a).n]$ = λ

and

$[(a χ u).n]/[(a χ b).n]$ = μ

(In any circumstance when I have omitted the underlining it is for convenience purposes, a, b, n and u are still vectors.)


2. The vectors a and b form a set of basis vectors for P₁ (A plane spanned by vectors a and b). However it is often desirable to use a set of orthogonal basis vectors.

Construct a vector b' that is in P ₁ and orthogonal to a.

Homework Equations

We are told that a and b are vectors in ℝ³, and that they are not parallel or anti-parallel. n is the normal vector to plane P₁.

The Attempt at a Solution

1. n= a x b

a x b = a x (aλ x bμ)
= a x bμ
= nμ

∴$(a χ u)/n$ = μ
$(a χ u)/(a χ b)$ = μ

But of course, you can't divide by a vector. What you do have correct is$$
a\times u =\mu(a\times b)$$with vectors on both sides. You can dot both sides of that equation with $n$, giving$$
(a\times u)\cdot n = \mu(a\times b) \cdot n$$Now this is a scalar equation which you can solve for $\mu$. Do you see why $(a\times b) \cdot n\ne 0\$, allowing you to divide by it?

And you do the similar thing to solve for $\lambda$.

2. I can't find any sort of methodology that would yield a vector, b', perpendicular to a in P₁ when only given the information above. I would assume it has something to do with question 1 (I could be wrong), though I can't see how they relate to one another.

Thanks in advance for any help.

For 2 think about where the vector $(a \times b)\times a$ would be.