Exercises

01

Parallelogram identity

Let $\mathbf{u}=(4,0,3)$ and $\mathbf{v}=(-2,1,5)$. Compute $|\mathbf{u}-\mathbf{v}|$ and $|\mathbf{u}+\mathbf{v}|$ and $|\mathbf{u}|$ and $|\mathbf{v}|$. Show that:

$$|\mathbf{u}+\mathbf{v}{|}^2+|\mathbf{u}-\mathbf{v}{|}^2=2|\mathbf{u}{|}^2+2|\mathbf{v}{|}^2$$

02

Angle between vectors

Find the angle between $(3,6,2)$ and $(4,2,4)$.

03

Orthogonality by hand

Determine which pairs of $\mathbf{u}=(2,-1,1)$ and $\mathbf{v}=(2,6,1)$ and $\mathbf{w}=(-4,1,2)$ are orthogonal.

04

Acute, orthogonal, or obtuse?

Are the angles between $\mathbf{u}=(1/2,1/2,5)$ and $\mathbf{v}=(3,1,-2)$ and $\mathbf{w}=(4,-3,0)$ obtuse or acute?

05

Dot product via unit vectors

Compute the dot product of $4{\mathbf{e}}_{\mathbf{i}}-3{\mathbf{e}}_{\mathbf{j}}$ and ${\mathbf{e}}_{\mathbf{i}}+2{\mathbf{e}}_{\mathbf{j}}+{\mathbf{e}}_{\mathbf{k}}$.

06

Projection by hand

Let $\mathbf{u}=(5,1,-3)$ and $\mathbf{v}=(4,4,2)$.

• (a) Find the projection of $\mathbf{u}$ to the direction of $\mathbf{v}$.
• (b) Write the vector $\mathbf{u}$ in terms of parts parallel and perpendicular to $\mathbf{v}$.

07

Cauchy-Schwarz

Prove the Cauchy-Schwarz inequality using the angle formula.

08

Cauchy-Schwarz $\Rightarrow$ Triangle

Prove the Triangle inequality using the Cauchy-Schwarz inequality and an expansion of $|\mathbf{u}+\mathbf{v}{|}^2$ with the dot product.

09

Orthogonality identity

Suppose that vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $|\mathbf{u}+\mathbf{v}{|}^2=|\mathbf{u}{|}^2+|\mathbf{v}{|}^2$. Show that they must be orthogonal.

10

Cross products by hand

Let $\mathbf{u}=(1,-2,1)$ and $\mathbf{v}=(3,1,-2)$. Compute in coordinates:

• (a) $\mathbf{u}\times \mathbf{v}$
• (b) $\mathbf{v}\times \mathbf{u}$
• (c) $\mathbf{u}\times \mathbf{u}$

11

Cross product $⇝$ left-handed system

Use the cross product to find a unit vector $\mathbf{v}$ that is perpendicular to both $\mathbf{a}=(1,-4,1)$ and $\mathbf{b}=(2,3,0)$ such that the triple $(\mathbf{a},\mathbf{v},\mathbf{b})$ forms a left-handed system.

12

Cross product via unit vectors

Compute the cross product of $4{\mathbf{e}}_{\mathbf{i}}-3{\mathbf{e}}_{\mathbf{j}}$ and ${\mathbf{e}}_{\mathbf{i}}+2{\mathbf{e}}_{\mathbf{j}}+{\mathbf{e}}_{\mathbf{k}}$ using the unit vector cross products.

13

Jacobi identity

Use the Lagrange triple product identity to prove the Jacobi identity:

$$\mathbf{a}\times (\mathbf{b}\times \mathbf{c})+\mathbf{b}\times (\mathbf{c}\times \mathbf{a})+\mathbf{c}\times (\mathbf{a}\times \mathbf{b})=0$$