Using the definition of the scalar product, find the angle between the following vectors. (find the smallest angle.)
a) a = 5i - 9j and b = 5i - 5j
(b) a = -8i + 5j and b = 3i - 4j + 2k
(c) a= -2i + 2j and b = 3j + 4k

How does the image distance (di) of a convex lens compare with the image distance of a concave lens? a. the image distance of the convex lens is positive, and that of the concave lens is negative. b. both are negative for a virtual image. c. both are positive for a virtual image d. the image distance of the convex lens is negative, and that of the concave lens is positive.

Sawyer is studying diffraction. he draws a diagram of a plane wave to show how light waves travel. which best describes sawyer’s error? the wave fronts should be perpendicular to the direction in which the waves move. the arrow showing the direction of movement of the waves should be pointing to the left. the arrow showing the direction of movement of the waves should be pointing downward. the wave fronts should be both parallel and perpendicular to the direction in which the waves move.

An ideal otto cycle has a compression ratio of 10.5, takes in air at 90 kpa and 40°c, and is repeated 2500 times per minute. using constant specific heats at room temperature, determine the thermal efficiency of this cycle and the rate of heat input if the cycle is to produce 90 kw of power.