91Ó°ÊÓ

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\vec{i}, \quad \vec{v}=\vec{j}\)

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(-2,5)\), parallel to \(\vec{d}=\langle 0,1\rangle\).

Give the equation of the described plane in standard and general forms. Contains the point (2,-6,1) and the line $$ \ell(t)=\left\\{\begin{array}{l} x=2+5 t \\ y=2+2 t \\ z=-1+2 t \end{array}\right. $$

In Exercises 15-22, determine if the described lines are the same line, parallel lines, intersecting or skew lines. If intersecting, give the point of intersection. $$ \text { } \begin{aligned} \vec{\ell}_{1}(t) &=\langle 1,2,1\rangle+t\langle 2,-1,1\rangle \\ \vec{\ell}_{2}(t) &=\langle 3,3,3\rangle+t\langle-4,2,-2\rangle \end{aligned} $$

Find the measure of the angle between the two vectors in both radians and degrees. \(\vec{u}=\langle 8,1,-4\rangle, \vec{v}=\langle 2,2,0\rangle\)

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\vec{i}, \quad \vec{v}=\vec{k}\)

Give the equation of the described plane in standard and general forms. Contains the point (5,7,3) and the line $$ \ell(t)=\left\\{\begin{array}{l} x=t \\ y=t \\ z=t \end{array}\right. $$

Sketch the cylinder in space. \(y=\cos z\)

Find the measure of the angle between the two vectors in both radians and degrees. \(\vec{u}=\langle 1,7,2\rangle, \vec{v}=\langle 4,-2,5\rangle\)

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\vec{j}, \quad \vec{v}=\vec{k}\)