Assignment# 1 Ex: 12.1 Q.1. Find the equations of two spheres that are centered at the origin and are tangent to the sphere of radius 1 centered at (3, −2, 4). Q.2. Describe the surface whose equation is x2 + y 2 + z 2 − 3x + 4y − 8z + 25 = 0. Ex: 12.2 √ Q.3. Find the vectors that has length 17 and same direction as h7, 0, −6i. Q.4. Find the component form of the vector ~v in 2-space that has the length k~v k = 5 and makes the angle θ = 5π with the positive x-axis. 6 Ex: 12.3 Q.5. Find r so that the vector from the point A(1, −1, 3) to the point B(3, 0, 5) is orthogonal 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 dimensions 10cm by 15cm by 25cm makes with the edges of the box. Q.7. Find component of ~v = −3ˆi − 2ˆj along ~b = 2ˆi + ˆj and the vector component of ~v orthogonal 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) to the line through A(1, 1, 0) and B(−2, 3, −4). Ex: 12.4 Q.9. Find two unit vectors that are parallel to the yz-plane and are orthogonal to the ˆ vector 3ˆi − ˆj + 2k. Q.10. Find the area of the triangle with vertices P (2, 0, −3), Q(1, 4, 5), R(7, 2, 9). ˆ ~v = ˆi + ˆj + 2k. ˆ Q.11. Consider the parallelepiped with adjacent edges ~u = 3ˆi + 2ˆj + k, 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: 12.5 Q.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) and intersects 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 passes through their point of intersection.
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