91Ó°ÊÓ

Use the techniques of Problems 9 and 10 to find the vector or point. Find the position vector of the point \(\frac{1}{3}\) of the way from \(A(2,7)\) to \(B(14,5)\).

Use the techniques of Problems 9 and 10 to find the vector or point. Find the position vector of the point \(\frac{2}{3}\) of the way from \(C(11,5)\) to \(D(2,17)\).

Find a unit vector in the direction from the first point to the second point, and write its direction cosines. $$(-5,3,2) \text { to }(3,2,6)$$

Find a vector equation of the line from the first point to the second. $$(6,-2,7) \text { to }(10,6,26)$$

The cross product of the normal vectors to two planes is a vector that points in the direction of the line of intersection of the planes. Find a particular equation of the plane containing (-3,6,5) and normal to the line of intersection of the planes \(3 x+5 y+4 z=-13\) and \(6 x-2 y+7 z=8\).

Find the position vector of the indicated point. \(\frac{2}{3}\) of the way from (7,8,11) to (34,32,14)

Use the techniques of Problems 9 and 10 to find the vector or point. Find the midpoint of the segment connecting \(E(6,2)\) and \(F(10,-4) .\) From the result, give a quick way to find the midpoint of the segment connecting two given points.

Prove that the vector is a unit vector, and find its direction cosines and direction angles. $$\frac{7}{9} \vec{i}+\frac{4}{9} \vec{j}-\frac{4}{9} \vec{k}$$

Find \(\vec{a} \cdot \vec{b}\) and the angle between \(\vec{a}\) and \(\vec{b}\) when they are tail-to-tail. $$\begin{aligned} &\vec{a}=2 \vec{\imath}+5 \vec{j}+3 \vec{k}\\\ &\vec{b}=7 \vec{\imath}-\vec{j}+4 \vec{k} \end{aligned}$$

Prove that these two planes are perpendicular. $$ \begin{array}{l} 2 x-5 y+3 z=10 \\ 7 x+4 y+2 z=17 \end{array} $$