The coordinates of four adjacent vertices (from one vertex, edges extend to the others) of a parallelepiped are (1,2,3), (0,0,0), (2,1,0), and (2,4,2).

What is the volume of the parallelepiped defined by these points?

Give the answer to one decimal place.

What is the area of the triangle with vertices A(1,1,1), B(1,2,3), and C(3,2,1)?

Give the answer to two decimal places.

What is the area of the parallelogram determined by the vectors = [1, 2, 3] and = [3, 2, 1]?

Give the answer as a real number to three decimal places.

The height of the door is 2250 mm and its width is 1000 mm. The door is opened by 40°. What is the angle formed by the diagonal of the door (i.e., the diagonal of the rectangle formed by the door) with itself in its initial and final positions?

Give the answer to the nearest degree.

A similar problem was in the section "Right Triangle," but now that vectors are mastered, this problem can be solved more efficiently using vectors.

It is recommended to choose the coordinate system so that the origin is at the bottom corner on the hinge side, the initial direction of the door is chosen as the direction of the X-axis, and the door opens in the Y-direction. Z is, of course, upwards.

What is the angle between the vectors and ?

Give the answer to one decimal place.

Calculate the dot product of the vectors [-1, 0, 3] and [-2, 3, 2]. Give the answer as an integer.

The vectors and are perpendicular to each other. and .

Determine

Give the answer to three decimal places.

Determine when and and the angle between the vectors is 115°.

Give the answer to one decimal place.

Given vectors and , it is known that , , and . Determine the angle between and in degrees, accurate to one decimal place.

The x-coordinate of the vector [-10, 5, -2] is increased by 5.

How much does the length of the vector change? Indicate an increase as positive and a decrease as negative.

Give your answer to one decimal place.