Exercises: Difference between revisions
Eric Lengyel (talk | contribs) (Created page with "These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated'']. == Exercises for Chapter 2 == '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$. '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective alg...") |
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'''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product. | '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product. | ||
'''3.''' Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights. | |||
'''4.''' Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$. | |||
Revision as of 01:53, 29 April 2024
These are exercises accompanying the book Projective Geometric Algebra Illuminated.
Exercises for Chapter 2
1. Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.
2. Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.
3. Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.
4. Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.