91Ó°ÊÓ

State whether the lines are parallel or perpendicular and find the angle between the lines. $$\left\\{\begin{array}{ll} x=3-t & \\ y=4 & \text { and } \\ z=-2+2 t & \end{array}\left\\{\begin{array}{l} x=1+2 s \\ y=7-3 s \\ z=-3+s \end{array}\right.\right.$$

(a) find two unit vectors parallel to the given vector and (b) write the given vector as the product of its magnitude and a unit vector. $$4 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}$$

Find the vector with initial point \(A\) and terminal point \(\boldsymbol{B}.\) $$A=(4,3), B=(1,0)$$

Find two unit vectors orthogonal to the two given vectors. $$\mathbf{a}=-2 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k}, \mathbf{b}=2 \mathbf{i}-\mathbf{k}$$

Determine whether the lines are parallel, skew or intersect. $$\left\\{\begin{array}{ll} x=4+t & \\ y=2 & \text { and } \\ z=3+2 t & \end{array}\left\\{\begin{array}{l} x=2+2 s \\ y=2 s \\ z=-1+4 s \end{array}\right.\right.$$

(a) find two unit vectors parallel to the given vector and (b) write the given vector as the product of its magnitude and a unit vector. $$\text { From }(1,2,3) \text { to }(3,2,1)$$

Sketch the appropriate traces, and then sketch and identify the surface. $$z=4-x^{2}-y^{2}$$

Use the cross product to determine the angle between the vectors, assuming that \(0 \leq \theta \leq \frac{\pi}{2}\). $$\mathbf{a}=\langle 1,0,4\rangle, \mathbf{b}=\langle 2,0,1\rangle$$

Find the vector with initial point \(A\) and terminal point \(\boldsymbol{B}.\) $$A=(-1,2), B=(1,-1)$$

Determine whether the lines are parallel, skew or intersect.$$\left\\{\begin{array}{l} x=3+t \\ y=3+3 t \\ z=4-t \end{array} \quad \text { and } \quad\left\\{\begin{array}{l} x=2-s \\ y=1-2 s \\ z=6+2 s \end{array}\right.\right.$$