91Ó°ÊÓ

Vector Projection of u onto \(\mathbf{v} \quad\) (a) Calculate proj, \(\mathbf{u}\). (b) Resolve 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 \(\mathbf{v}\) $$\mathbf{u}=\langle 1,2\rangle, \quad \mathbf{v}=\langle 1,-3\rangle$$

Find all solutions of the given equation. $$3 \tan ^{2} \theta-1=0$$

Three vectors u, v, and w are given. (a) Find their scalar triple product \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$\mathbf{u}=\langle 2,3,-2\rangle, \quad \mathbf{v}=\langle- 1,4,0\rangle, \quad \mathbf{w}=\langle 3,-1,3\rangle$$

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 \(\mathbf{v}.\) $$\mathbf{u}=\langle 2,7\rangle, \quad \mathbf{v}=\langle 3,1\rangle$$

Determine whether or not the given vectors are perpendicular. $$\langle 0.3,1.2,-0.9\rangle, \quad\langle 10,-5,10\rangle$$

Determine whether or not the given vectors are perpendicular. $$\langle x,-2 x, 3 x\rangle, \quad\langle 5,7,3\rangle$$

Find all solutions of the given equation. $$\cot \theta+1=0$$

Vector Projection of u onto \(\mathbf{v} \quad\) (a) Calculate proj, \(\mathbf{u}\). (b) Resolve 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 \(\mathbf{v}\) $$\mathbf{u}=\langle 11,3\rangle, \quad \mathbf{v}=\langle- 3,-2\rangle$$

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

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 \(\mathbf{v}.\) $$\mathbf{u}=\langle- 2,5\rangle, \quad \mathbf{v}=\langle 2,-8\rangle$$