By Walsh J.L.

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**Example text**

Given the vector v = ofv. J4+9 = SOLUTION. We have Ivl = u 2i - + 3j, find a unit vector in the direction JT3. The desired vector u is I = JT3v = - 2. JT31 3. + JT3 J. EXAMPLE 4. Given the vector v = 2i - 4j. Find the directed line segment has coordinates (3, - 5). AS of v, given that A SOLUTION. Denote the coordinates of B by X B , YB' Then we have (by Theorem 2) XB - Therefore x B = 5, YB = 3 =2 -9. and YB +5= -4. 44 2. 8 of v have the given coordinates. Draw a figure. I. A(3, -2), B(I,5) 2. A( -4, I), B(2, -I) 3.

The next example shows how to transform one representation into another. EXAMPLE 2. The two planes 2x + 3y - 4z - 6 =0 and 3x - y + 2z + 4 = 0 intersect in a line. ) Find a set of parametric equations of the line of intersection. SOLUTION. t, Z = t, which are the desired parametric equations. Three planes may be parallel, may pass through a common line, may have no common points, or may have a unique point of intersection. If they have a unique point of intersection, the intersection point may be found by solving simultaneously the three equations of the planes.

P[(2, -I, -2); L[ : x = 2 + 3t, y = 0, 12. p[ ( - I, 2, - 3); L, : x = - I + 5t, Y = + 3t, Z Z Z = I = = -4t I + 3t -1 - 2t + 2t, Z = -I + 3t In each of Problems 13 through 16, find the equations of the line through PI and perpendicular to the given plane MI' + 3y + Z - 3 = 0 +4=0 + 2z + 3 = 0 13. PI (-2,3, I); M I : 2x 14. PI(I, -2, -3); MI 15. P I ( -1,0, -2); M[: x 16. P[(2, -I, -3); M I :x=4 : 3x - y - 2z In each of Problems 17 through 20, find an equation of the plane through PI and parallel to the plane 11>.