# Exercises

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 $$\mathbf m$$.

**5.** Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.

**6.** Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605})^\unicode["segoe ui symbol"]{x2606}$$.

**7.** Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode["segoe ui symbol"]{x2605} = \mathbf a^\unicode["segoe ui symbol"]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.

**8.** Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.

**9.** Derive the relationship between left and right interior products shown in Equation (2.110).

**10.** Derive Equation (2.159), which is the expansion analog of Equation (2.129).

**11.** Derive a formula for $$\mathbf u^{\unicode["segoe ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.

**12.** Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.