91Ó°ÊÓ

Sketch the surfaces HYPERBOLOIDS $$y^{2}+z^{2}-x^{2}=1$$

Find the point of intersection of the lines \(x=t, y=-t+2\) \(z=t+1,\) and \(x=2 s+2, y=s+3, z=5 s+6,\) and then find the plane determined by these lines.

Which of the following are always true, and which are not always true? Give reasons for your answers. a. \(\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}\) b. \(\mathbf{u} \times \mathbf{v}=-(\mathbf{v} \times \mathbf{u})\) c. \((-\mathbf{u}) \times \mathbf{v}=-(\mathbf{u} \times \mathbf{v})\) d. \((c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v}) \quad \text { (any number } c)\) e. \(c(\mathbf{u} \times \mathbf{v})=(c \mathbf{u}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v}) \quad \text { (any number } c)\) f. \(\mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|^{2}\) g. \((\mathbf{u} \times \mathbf{u}) \cdot \mathbf{u}=0\) h. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u}=\mathbf{v} \cdot(\mathbf{u} \times \mathbf{v})\)

Express each vector as a product of its length and direction. $$\frac{3}{5} \mathbf{i}+\frac{4}{5} \mathbf{k}$$

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(3,4,5), \quad P_{2}(2,3,4)$$

Express each vector as a product of its length and direction. $$\frac{1}{\sqrt{6}} \mathbf{i}-\frac{1}{\sqrt{6}} \mathbf{j}-\frac{1}{\sqrt{6}} \mathbf{k}$$

Sketch the surfaces HYPERBOLOIDS $$z^{2}-x^{2}-y^{2}=1$$

Find the plane containing the intersecting lines. \(\begin{array}{ll}L 1: x=-1+t, & y=2+t, z=1-t ; \quad-\infty

Orthogonal unit vectors If \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are orthogonal unit vectors and \(\mathbf{v}=a \mathbf{u}_{1}+b \mathbf{u}_{2},\) find \(\mathbf{v} \cdot \mathbf{u}_{1}\)

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(0,0,0), \quad P_{2}(2,-2,-2)$$