Assignment# 1

Ex: 12.1Q.1. Find the equations of two spheres that are centered at the origin and are tangentto the sphere of radius 1 centered at (3,−2, 4).Q.2. Describe the surface whose equation is x2 + y2 + z2 − 3x+ 4y − 8z + 25 = 0.

Ex: 12.2Q.3. Find the vectors that has length

√17 and same direction as 〈7, 0,−6〉.

Q.4. Find the component form of the vector ~v in 2-space that has the length ‖~v‖ = 5and makes the angle θ = 5π

6with the positive x-axis.

Ex: 12.3Q.5. Find r so that the vector from the point A(1,−1, 3) to the point B(3, 0, 5) is or-thogonal to the vector from A to the point P (r, r, r).Q.6. Find, to the nearest degree, the angles that a diagonal of a box with dimensions10cm by 15cm by 25cm makes with the edges of the box.Q.7. Find component of ~v = −3i − 2j along ~b = 2i + j and the vector component of ~vorthogonal to ~b. Then sketch the vectors ~v, proj~b~v, and ~v − proj~b~v.Q.8. Use the method of Dot Product to find the distance from the point P (−3, 1, 2) tothe line through A(1, 1, 0) and B(−2, 3,−4).

Ex: 12.4Q.9. Find two unit vectors that are parallel to the yz-plane and are orthogonal to thevector 3i− j + 2k.Q.10. Find the area of the triangle with vertices P (2, 0,−3), Q(1, 4, 5), R(7, 2, 9).Q.11. Consider the parallelepiped with adjacent edges ~u = 3i + 2j + k, ~v = i + j + 2k.Find the volume. Find the area of face determined by ~u and ~w. Find the angle between~u and the plane containing the face determined by ~v and ~w.Q.12. Use the method of Cross Product to find the distance from the point P (−3, 1, 2)to the line through A(1, 1, 0) and B(−2, 3,−4).

Ex: 12.5Q.13. Show that the lines L1 : x = 2 + 8t, y = 6 − 8t, z = 10t and L2 : x = 3 + 8t,y = 5− 3t, z = 6 + t are skew.Q.14. Show that the lines L1 : x = 1 + 3t, y = −2 + t, z = 2t and L2 : x = 4 − 6t,y = −1− 2t, z = 2− 4t are the same.Q.15. Show that the lines L1 : x = 2− t, y = 2t, z = 1+ t and L2 : x = 1+2t, y = 3−4t,z = 5− 2t are parallel, and find the distance between them.Q.16. Find the parametric equations of the line that contains the point P (0, 2, 1) andintersects the line L : x = 2t, y = 1− t, z = 2 + t, at a right angle.Q.17. Let L1 and L2 be the lines whose parametric equations are L1 : x = 4t, y = 1−2t,z = 2 + 2t, L1 : x = 1 + t, y = 1− t, z = −1 + 4t. Show that lines intersect at the point(2, 0, 3). Find, to the nearest degree, the acute angle between them at their intersection.Find also parametric equations for the line that is perpendicular to L1 and L2 and passesthrough their point of intersection.

Assignment# 1

Ex: 12.6Q.18. Find the equation of the plane through (1, 2,−1) that is perpendicular to the lineof intersection of the planes 2x+ y + z = 2 and x+ 2y + z = 3.Q.19. Find the equation of the plane that contains the point (2, 0, 3) and the line x =−1 + t, y = t, z = −4 + 2t.Q.20. Show that the lines x = −2 + t, y = 3 + 2t, z = 4− t and x = 3− t, y = 4− 2t,z = t are parallel and find an equation of the plane they determine.Q.21. Find the distance between the skew lines x = 3 − t, y = 4 + 4t, z = 1 + 2t andx = t, y = 3, z = 2t.

Ex: 12.7Q.22. Identify and sketch the quadric surface 9x2 + 4y2 − 36z2 = 0.Q.23. Identify and sketch the quadric surface 16z = y2 − x2.Q.24. Use the ellipsoid 4x2 + 9y2 + 18z2 = 72. To find an equation of the elliptical tracein the plane z =

√2. Find the lengths of the major and minor axes of the ellipse. Find

the coordinates of the foci of the ellipse. Describe the orientation of the focal axis ofellipse relative to the coordinate axes.Q.25. Use the hyperbolic paraboloid z = y2 − x2. To find an equation of the hyperbolictrace in the plane z = 4. Find the vertices of the hyperbola. Find the coordinates of thefoci of the hyperbola. Describe the orientation of the focal axis of hyperbola relative tothe coordinate axes.

Ex: 12.8Q.26. Convert (4

√3, 4,−4) from rectangular to cylindrical and spherical coordinates.

Q.27. Convert (4, π/6, 3) from cylindrical to rectangular and spherical coordinates.Q.28. Convert (5, π/6, π/4) from spherical to rectangular and cylindrical coordinates.Q.29. ρ sinφ = 2 cos θ is an equation in spherical coordinates. Express the equation inrectangular coordinates and sketch the graph.Q.30. x2 + y2 − 6y = 0 is an equation of a surface in rectangular coordinates. Find anequation of the surface in cylindrical coordinates and spherical coordinates.