Vectors

Question 1

The magnitudes of displacements a and b are 3 m and 4 m, respectively, and c = a + b. Considering various orientations of a and b, what are (a) the maximum possible magnitude for c and (b) the minimum possible magnitude?

Answer 1

Magnitude ε R⁺: |c| = (c_x² + c_y²)^1/2

(a) Maximum of c is | (3,0) + (4,0) | = | (0,-3) + (0,-4) | = ….. = |7| = |-7| = 7, this is assuming that these vectors and meet on the same line, which is the only possible way to reach a maximum magnitude.
(b) Minimum of c is acheived by having antiparallel vectors, thus we can have either of the combinations; | (-3,0) + (4,0) | = | (3,0) + (-4,0) | = | (0,-3) + (0,4) | = …. = 1.

Question 2

Which type of product does the commutatuive law not apply to?

Answer 2

Vector product.

b x a = -(a x b).

Question 3

A ship set out to sail to a point 120 km due north. An unexpected storm blows the ship to a point 100 km due east of its starting point. (a) How far and (b) in what direction must it now sail to reach its original destination?

Answer 3

(a) = (120² + 110² )^1/2

(b) Ø = tan^-1(120/100)