Vector is 5.0 cm long and vector is 4.0 cm long. Find the angle between these two vectors when and .
What is the component of the force vector along the force vector ?
The following figure shows a triangle formed by the three vectors , , and . If vector is drawn between the midpoints of vectors and , show that .
Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle to become a new coordinate system , as shown in the following figure. A point in a plane has coordinates (x, y) in S and coordinates in .
(a) Show that, during the transformation of rotation, the coordinates in are expressed in terms of the coordinates in S by the following relations:
(b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that
(c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that