91Ó°ÊÓ

For the following exercises, point \(P\) and vector \(\mathbf{v}\) are given. Let \(L\) be the line passing through point \(P\) with direction \(\mathbf{v}\). Find parametric equations of line \(L\). Find symmetric equations of line \(L\). Find the intersection of the line with the \(x y\) -plane. \(P(3,1,5), \mathbf{v}=\langle 1,1,1\rangle\)

For the following exercises, line \(L\) is given. Find point \(P\) that belongs to the line and direction vector \(\mathbf{v}\) of the line. Express \(\mathbf{v}\) in component form. Find the distance from the origin to line \(L\). \(x=1+t, y=3+t, z=5+4 t, t \in \mathbb{R}\)

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.\(y=2 x^{2}\)

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

A vector \(\mathbf{v}\) has initial point \((-2,5)\) and terminal point \((3,-1)\). Find the unit vector in the direction of \(\mathbf{v}\). Express the answer in component form.

The vector \(\mathbf{v}\) has initial point \(P(1,1)\) and terminal point \(Q\) that is on the \(x\) -axis and left of the initial point. Find the coordinates of terminal point \(Q\) such that the magnitude of the vector \(\mathbf{v}\) is \(\sqrt{10}\)

Let \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) be three-dimensional vectors and \(c\) be a real number. Prove the following properties of the cross product. a. \(\mathbf{u} \times \mathbf{u}=0\) b. \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w})\) c. \(c(\mathbf{u} \times \mathbf{v})=(\mathrm{cu}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v})\) d. \(\mathbf{u} \cdot(\mathbf{u} \times \mathbf{v})=\mathbf{0}\)

For the following exercises, the spherical coordinates \((\rho, \theta, \varphi)\) of a point are given. Find the rectangular coordinates \((x, y, z)\) of the point.\(\left(3, \frac{\pi}{4}, \frac{\pi}{6}\right)\)

Nonzero vectors \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers \(\alpha\) and \(\beta\) such that \(\mathbf{w}=\alpha \mathbf{u}+\beta \mathbf{v}\). Otherwise, the vectors are called linearly independent. Show that \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) are coplanar if and only if they are linear dependent.

Write the standard form of the equation of the ellipsoid centered at the origin that passes through points \(A(2,0,0), B(0,0,1)\), and \(C\left(\frac{1}{2}, \sqrt{11}, \frac{1}{2}\right)\)