내적과 정사영

Use vectors to show that a parallelogram with equal diagonals is a rectangle.

Use vectors to show that the diagonals of a rhombus are perpendicular.

Show that $\text{u}\cdot (\text{v}+\text{w})=\text{u}\cdot \text{v}+\text{u}\cdot \text{w}$ is true for any vectors $\text{u}$, $\text{v}$, and $\text{w}$.

Consider $\text{u}=\langle a,b,c\rangle$ a nonzero three-dimensional vector. Let $\cos \,\alpha ,$ $\cos \,\beta ,$ and $\cos \,\gamma$ be the direction cosines of $\text{u}$. Show that ${\cos }^{2}\alpha +{\cos }^{2}\beta +{\cos }^{2}\gamma =1.$

[T] Consider $\text{r}(t)=\langle \cos \,t,\sin \,t,2t\rangle$ the position vector of a particle at time $t\in [0,30],$ where the components of r are expressed in centimeters and time in seconds. Let $\vec{OP}$ be the position vector of the particle after 1 sec.

a. Show that all vectors $\vec{PQ},$ where $Q(x,y,z)$ is an arbitrary point, orthogonal to the instantaneous velocity vector $\text{v}(1)$ of the particle after 1 sec, can be expressed as $\vec{PQ}=\langle x-\cos \,1,y-\sin \,1,z-2\rangle ,$ where $x\,\sin \,1-y\,\cos \,1-2z+4=0.$ The set of point Q describes a plane called the normal plane to the path of the particle at point P. b. Use a CAS to visualize the instantaneous velocity vector and the normal plane at point P along with the path of the particle.

Create your own vectors $\vec u$, $\vec v$ and $\vec w$ in $\mathbb{R}^2$ and show that $\vec u\cdot (\vec v+\vec w) = \vec u\cdot \vec v + \vec u\cdot \vec w$.

Create your own vectors $\vec u$ and $\vec v$ in $\mathbb{R}^3$ and scalar $c$ and show that $c(\vec u\cdot \vec v) = \vec u\cdot (c\vec v)$.