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 $|\mathbf{v}|$. Verify 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

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?

04

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 $\mathbf{v}$.

05

Cauchy-Schwarz

Prove the Cauchy-Schwarz inequality using the angle formula.

06

Cauchy-Schwarz $\Rightarrow$ Triangle

Prove the Triangle inequality using the Cauchy-Schwarz inequality by “foiling” $|\mathbf{u}+\mathbf{v}{|}^2$ after writing it using the dot product.

07

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. (Hint: use the “foiling” trick in the previous exercise.)

08

Line and plane intersection

Find the point at which the plane $3x-9y+2z-7=0$ and the line $\mathbf{r}(t)=(1,2,1)+t(-2,0,1)$ intersect.

09

Distance between parallel planes

Find the distance from the plane given by $15x+3y-3z-10=0$ to the parallel plane given by $5x+y-z-3=0$.