Several unit vectors r s t u n and e in the xyplane not thre

Several unit vectors r, s, t u, n, and e in the xy-plane (not three-dimensional space) are shown in the figure. Using the geometric definition of the dot product, are the following dot products positive, negative, or zero? You may assume that angles that look the same are the same.

Solution

If the angle between two vectors is

(i) less than 90 degrees then their dot product is positive.

(ii) equal to 90 degrees, then their dot product is 0

(iii) greater than 90 degrees then their dot product is positive.

Now let us answer.

(1) e.r is negative (as the angle between them is greater than 90 degrees)

(2) e.s = 0 ( as the angle between them is 90 degrees)

(3) n.e = 0 ( as the angle between them is 90 degrees)

(4) s.t is positive (as the angle between them is less than 90 degrees)

(5) r. s is positive (as the angle between them is less than 90 degrees)

(6) n.t is negative (as the angle between them is greater than 90 degrees)

(7) t.u is positive (as the angle between them is less than 90 degrees)

(8) r.u is negative (as the angle between them is greater than 90 degrees).