Problems

In Problems l-ll, (a) compute A · B, (b) test whether A is perpendicular or parallel to B, and (c) find the cosine of the angle between A and B using A · B.

13            Find the cost of the commodity vector A = 15i + 4j + 6k at the price vector P = i + 2j + 3k.

14            Find the profit or loss if a trader buys the commodity vector A = 3i + 16j + 4k at the price vector P = 2i + 4j + 6k and sells it at the price vector 3i + 2j + 10k.

15            A trader initially has the commodity vector A = i + 3j + 6k. He sells his whole commodity vector at the price P = 3i + j + 2k and uses the revenue from this sale to buy an equal amount of each commodity. Find his new commodity vector.

16            Find the amount of work done by the force vector F = 3i - j - 4k acting along the displacement vector S = 5i + 3j + k.

17            Find the work done by a force vector of magnitude 10 acting along a displacement of length 40 if the angle between the force and displacement is 45°.

18            Prove that the basis vectors i, j, k are perpendicular.

19            Find a vector in the plane perpendicular to A = i + j.

20            Find a vector in the plane perpendicular to A = 2i - 9j.

21             Compute A × B where A = i - 3j + k, B = -i - j + k.

22            Compute A × B where A = i - j + k, B = i + j + k.

23            Compute A × B where A = i + k, B = j - k.

24            Find a vector perpendicular to both A = i + j - k, B = i - j + k.

25            Find a vector perpendicular to both A = i + 2j + 3k, B = i + 3j + 4k.

26            Find a vector perpendicular to both A = -i - 4j + k, B = j - 2k.

27            Find a unit vector in the plane perpendicular to A = 3i - 4j.

28            Find a unit vector in the plane perpendicular to A = 2i - j.

29            Find a unit vector perpendicular to both A = i + j, B = k.

30            Find a unit vector perpendicular to both A = 2i + 3k, B = - i + j - k.

31             Find the angle between two long diagonals of a cube.

32            Find the angle between a long diagonal and a diagonal along a face of a cube.

33            Find the angle between the diagonals of two adjacent faces of a cube.

34            Show that the inner product of two unit vectors is equal to the cosine of the angle between them.

35            Use inner products to prove that the diagonals of a rhombus (a parallelogram whose sides have equal lengths) are perpendicular.

36            Prove the Distributive Law for vector products.

(sA + (B) × C = s(A × C) + t(B × C).

37            Prove the Anticommutative Law for vector products.

B × A = -(A × B).

38             Prove that A ∥ B if and only if A × B = 0 (where A, B are nonzero).

39            Show that the length of A × B is equal to the area of the parallelogram with sides A and B, in symbols

|A × B| = |A| |B|sin θ.

40             Prove that the "scalar triple product" A · (B × C) is equal to the volume of a parallelo-piped with edges A, B. and C.