91Ó°ÊÓ

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). \(P(3,2,2), \mathbf{n}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}\)

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

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

For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.[?] \(\rho=6 \csc \varphi \sec \theta\)

Consider vectors \(\mathbf{u}=2 \mathbf{i}+4 \mathbf{j}\) and \(\mathbf{v}=4 \mathbf{j}+2 \mathbf{k}\) a. Find the component form of vector \(\mathbf{w}=\operatorname{proj}_{\mathrm{u}} \mathbf{v}\) that represents the projection of \(\mathrm{v}\) onto \(\mathbf{u}\). b. Write the decomposition \(\mathbf{v}=\mathbf{w}+\mathbf{q}\) of vector \(\mathbf{v}\) into the orthogonal components \(\mathbf{w}\) and \(\mathbf{q}\), where \(\mathbf{w}\) is the projection of \(\mathrm{v}\) onto \(\mathrm{u}\) and \(\mathbf{q}\) is a vector orthogonal to the direction of \(\mathrm{u}\).

[T] A force \(\mathbf{F}\) of \(50 \mathrm{~N}\) acts on a particle in the direction of the vector \(\overrightarrow{O P}\), where \(P(3,4,0)\). a. Express the force as a vector in component form. b. Find the angle between force \(\mathbf{F}\) and the positive direction of the \(x\) -axis. Express the answer in degrees rounded to the nearest integer.

Find all vectors \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle\) that satisfy the equation \(\langle 1,1,1\rangle \times \mathbf{w}=\langle-1,-1,2\rangle\).

Show that quadric surface \(x^{2}+y^{2}+z^{2}+2 x y+2 x z+2 y z+x+y+z=0\) reduces to two parallel planes.

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n}\). \(P(1,2,3), \mathbf{n}=\langle 1,2,3\rangle\)

Find vector \(\mathbf{v}\) with the given magnitude and in the same direction as vector \(\mathbf{u}\). $$ \|\mathbf{v}\|=7, \mathbf{u}=\langle 3,4\rangle $$