벡터와 공간 전체 문제

For the following exercises, use the given vectors $\text{a}$ and $\text{b}.$

a. Determine the vector sum $\text{a}+\text{b}$ and express it in both the component form and by using the standard unit vectors. b. Find the vector difference $\text{a}-\text{b}$ and express it in both the component form and by using the standard unit vectors. c. Verify that the vectors $\text{a},$ $\text{b},$ and $\text{a}+\text{b},$ and, respectively, $\text{a},$ $\text{b},$ and $\text{a}-\text{b}$ satisfy the triangle inequality. d. Determine the vectors $2\text{a},$ $-\text{b},$ and $2\text{a}-\text{b}.$ Express the vectors in both the component form and by using standard unit vectors.

$\text{a}=2\text{i}+\text{j},$ $\text{b}=\text{i}+3\text{j}$

For the following exercises, use the given vectors $\text{a}$ and $\text{b}.$

a. Determine the vector sum $\text{a}+\text{b}$ and express it in both the component form and by using the standard unit vectors. b. Find the vector difference $\text{a}-\text{b}$ and express it in both the component form and by using the standard unit vectors. c. Verify that the vectors $\text{a},$ $\text{b},$ and $\text{a}+\text{b},$ and, respectively, $\text{a},$ $\text{b},$ and $\text{a}-\text{b}$ satisfy the triangle inequality. d. Determine the vectors $2\text{a},$ $-\text{b},$ and $2\text{a}-\text{b}.$ Express the vectors in both the component form and by using standard unit vectors.

$\text{a}=2\text{i},$ $\text{b}=-2\text{i}+2\text{j}$

Consider the vector $\text{a}(t)=\langle \cos \,t,\sin \,t\rangle$ with components that depend on a real number $t.$ As the number $t$ varies, the components of $\text{a}(t)$ change as well, depending on the functions that define them.

a. Write the vectors $\text{a}(0)$ and $\text{a}(\pi )$ in component form. b. Show that the magnitude $\| \text{a}(t)\|$ of vector $\text{a}(t)$ remains constant for any real number $t.$ c. As $t$ varies, show that the terminal point of vector $\text{a}(t)$ describes a circle centered at the origin of radius $1.$

Consider vector $\text{a}(x)=\langle x,\sqrt{1-{x}^{2}}\rangle$ with components that depend on a real number $x\in [-1,1].$ As the number $x$ varies*,* the components of $\text{a}(x)$ change as well, depending on the functions that define them.

a. Write the vectors $\text{a}(0)$ and $\text{a}(1)$ in component form. b. Show that the magnitude $\| \text{a}(x)\|$ of vector $\text{a}(x)$ remains constant for any real number $x$ c. As $x$ varies, show that if $\text{a}(x)$ is in standard position, then its terminal point describes a semicircle. Why is it only a semicircle?

Show that vectors $\text{a}(t)=\langle \cos \,t,\sin \,t\rangle$ and $\text{a}(x)=\langle x,\sqrt{1-{x}^{2}}\rangle$ are equivalent for $x=1$ and $t=2k\pi ,$ where $k$ is an integer.

Show that vectors $\text{a}(t)=\langle \cos \,t,\sin \,t\rangle$ and $\text{a}(x)=\langle x,\sqrt{1-{x}^{2}}\rangle$ are opposite for $x=1$ and $t=\pi +2k\pi ,$ where $k$ is an integer.

Consider $f$ and $g$ two functions defined on the same set of real numbers $D.$ Let $\text{a}=\langle x,f(x)\rangle$ and $\text{b}=\langle x,g(x)\rangle$ be two vectors that describe the graphs of the functions, where $x\in D.$ Show that if the graphs of the functions $f$ and $g$ do not intersect, then the vectors $\text{a}$ and $\text{b}$ are not equivalent.

Let $P(x,y,z)$ be a point situated at an equal distance from points $A(1,-1,0)$ and $B(-1,2,1).$ Show that point $P$ lies on the plane of equation $-2x+3y+z=2.$

Let $P(x,y,z)$ be a point situated at an equal distance from the origin and point $A(4,1,2).$ Show that the coordinates of point $P$ satisfy the equation $8x+2y+4z=21.$

The points $A,B,$ and $C$ are collinear (in this order) if the relation $\| \vec{AB}\| +\| \vec{BC}\| =\| \vec{AC}\|$ is satisfied. Show that $A(5,3,-1),$ $B(-5,-3,1),$ and $C(-15,-9,3)$ are collinear points.

For the following exercises, find vector $\text{u}$ with a magnitude that is given and satisfies the given conditions.

Show that points $A(1,0,1),$ $B(0,1,1),$ and $C(1,1,1)$ are not collinear.

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.

Show that $\text{u}\,\times \,\text{v}$ and $2\text{i}-14\text{j}+2\text{k}$ cannot be orthogonal for any $\alpha$ real number, where $\text{u}=\text{i}+7\text{j}-\text{k}$ and $\text{v}=\alpha \text{i}+5\text{j}+\text{k}.$

Show that $\text{u}\,\times \,\text{v}$ is orthogonal to $\text{u}+\text{v}$ and $\text{u}-\text{v},$ where $\text{u}$ and $\text{v}$ are nonzero vectors.

Show that $\text{v}\,\times \,\text{u}$ is orthogonal to $(\text{u}\cdot \text{v})(\text{u}+\text{v})+\text{u},$ where $\text{u}$ and $\text{v}$ are nonzero vectors.

Use the sine and cosine of the angle between two nonzero vectors $\text{u}$ and $\text{v}$ to prove Lagrange’s identity: ${\| \text{u}\,\times \,\text{v}\| }^{2}={\| \text{u}\| }^{2}{\| \text{v}\| }^{2}-{(\text{u}\cdot \text{v})}^{2}.$