1. The vector starts at the point and ends at the point (a) The component form of this vector is , (b) Compute the magnitude (c) The unit vector parallel to Note: to enter a vector 2. Suppose vectors but in the opposite direction has component form , type and in are given by Compute each of the following linear combinations of and : NOTE: To enter a vector , type " " (a) (b) (c) 3. Suppose vectors and in are given by (a) (b) Find a unit vector parallel to (and in the same direction as) 4. Suppose a vector Find a vector in : is given by that is parallel to (and in the same direction as) with magnitude 6: 5. A large container is being pulled across a surface using two ropes attached to pulleys. One rope pulls the container with a force of Newtons at an angle of the ground. The other rope pulls the container with a force of relative to the ground. Let by both ropes. relative to Newtons at an angle of be the combined force vector applied to the container (a) What is the magnitude of the combined force vector applied to the container by the box: (b) Find the component form of the total force vector ropes: applied to the container by the 6. The points , three vertices of a parallelogram. , and Two of the sides of the parallelogram are given by the vectors coordinates of the fourth vertex, , of this parallelogram. ,, Vectors 2 1. Suppose vectors and in are given by Compute each of the following: (a) (b) If is the angle between the vectors, (c) in and are . Find the 2. Suppose vectors and in are given by Compute each of the following: (a) (b) If is the angle between the vectors, (c) 3. Find the area of the triangle in , and 4. Suppose vectors whose vertices are , . and in Find a unit vector, are given by , with a positive -component that is orthogonal to BOTH and : 5. Suppose vectors , , and in are given by Find the volume of the parallelepiped spanned by the vectors , , and 6.Let be a constant. The points , , and vertices of a right triangle with right angle at the vertex in are the . What is the value of ? Planes 1. Find parametric equations that describe the line that passes through the points and . Note: Choose your parametrization so that the line passes through when when and 2. Find an equation for the plane that passes through the points , and , . Answer: 3. Let Let be the line in be the line in given by the parametric function given by the parametric function Find an equation for the plane that passes through the point to both lines and and is parallel . Answer: 4. Let Let be the line in given by the parametric function be the point of intersection of with the plane coordinates of ,, . Find the 5. Find the distance, , between the planes and 6. Let Let be the line given by the parametric function be the line given by the parametric equations Find the distance between these two lines. 7. Objects A and B are traveling in space. At time , Object A is located at the point given by At time , Object A is located at the point given by (a) Let be the point of intersection of these two lines. Find the coordinates of : ,, (b) Do the objects collide? Answer 'Yes' or 'No'. 8. Let Let be the plane in given by the equation be the plane passing through the points and , with a positive -component that is parallel to the line of intersection of the planes Let and . be the line given by the parametric function be the line given by the parametric equations Find the distance , . Find a unit vector, 9. Let . between these two lines. , \f\f