91Ó°ÊÓ

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is a unit vector, then \(a^{2}+b^{2}=1\).

Find the distance between the point and the line given by the set of parametric equations.\((-2,1,3) ; \quad x=1-t, \quad y=2+t, \quad z=-2 t\)

Find the distance between the point and the line given by the set of parametric equations.\((4,-1,5) ; \quad x=3, \quad y=1+3 t, \quad z=1+t\)

Convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates. \(4\left(x^{2}+y^{2}\right)=z^{2}\)

Sketch the vector \(v\) and write its component form. \(\mathrm{v}\) lies in the \(x z\) -plane, has magnitude 5 , and makes an angle of \(45^{\circ}\) with the positive \(z\) -axis.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(a=b\), then \(\|a \mathbf{i}+b \mathbf{j}\|=\sqrt{2} a\)

Convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates. \(x^{2}+y^{2}+z^{2}-2 z=0\)

In Exercises 99 and 100 , use vectors to find the point that lies two-thirds of the way from \(P\) to \(Q\). \(P(4,3,0), \quad Q(1,-3,3)\)

Verify that the lines are parallel, and find the distance between them.$$ \begin{aligned} &L_{1}: x=2-t, \quad y=3+2 t, \quad z=4+t \\ &L_{2}: x=3 t, \quad y=1-6 t, \quad z=4-3 t \end{aligned} $$

Verify that the lines are parallel, and find the distance between them.$$ \begin{aligned} &L_{1}: x=3+6 t, \quad y=-2+9 t, \quad z=1-12 t \\ &L_{2}: x=-1+4 t, \quad y=3+6 t, \quad z=-8 t \end{aligned} $$