91Ó°ÊÓ

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=\langle 1,-1,0\rangle, \quad \mathbf{b}=\langle- 1,0,1\rangle, \quad \mathbf{c}=\langle 0,-1,1\rangle $$

A description of a plane is given. Find an equation for the plane. The plane that crosses the \(x\) -axis where \(x=-2,\) the \(y\) -axis where \(y=-1,\) and the \(z\) -axis where \(z=3\)

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle 11,3\rangle, \quad \mathbf{v}=\langle- 3,-2\rangle $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\langle- 2,5\rangle, \quad \mathbf{v}=\langle 2,-8\rangle $$

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $$ \mathbf{u}=\langle 2,-2,-1\rangle, \quad \mathbf{v}=\langle 1,2,2\rangle $$

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle 2,9\rangle, \quad \mathbf{v}=\langle- 3,4\rangle $$

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\langle 0,-1\rangle, \quad \mathbf{v}=\langle- 2,0\rangle $$

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=\mathbf{i}-\mathbf{j}+\mathbf{k}, \quad \mathbf{b}=-\mathbf{j}+\mathbf{k}, \quad \mathbf{c}=\mathbf{i}+\mathbf{j}+\mathbf{k} $$

A description of a plane is given. Find an equation for the plane. The plane that is parallel to the plane \(x-2 y+4 z=6\) and contains the origin.

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{c}=6 \mathbf{i} $$