Question

Two boys searching for buried treasure are standing underneath the same tree. One boy walks east and then north. The other boy walks west and then north. Find the scalar product of their net displacements from the tree

Answer 1

Step-by-step explanation:

Initial Positions:

Initially, both boys start at the same tree, which we consider as the starting point (0,0) on a coordinate system.

First Boy's Displacement:

• The first boy moves first to the east (right) and then to the north (up).

• We represent his eastward movement as "d1" (a positive value) and his northward movement as "d2" (also a positive value).

• His combined or net displacement can be represented as the vector (d1, d2).

Second Boy's Displacement:

• The second boy moves first to the west (left) and then to the north (up).

• We represent his westward movement as "d3" (a positive value) and his northward movement as "d4" (also a positive value).

• His combined or net displacement can be represented as the vector (-d3, d4) because he's moving in the opposite direction of east.

Scalar Product of Net Displacements:

• To find the scalar product (dot product) of their net displacements, we use the formula: A⋅B = |A| * |B| * cos(θ), where A and B are vectors, |A| and |B| are their magnitudes, and θ is the angle between them.

• In this case, the angle θ between their net displacements is 90 degrees because one boy went east and the other went west.

• The cosine of 90 degrees, cos(90°), is 0.

• Therefore, the scalar product of their net displacements is |A| * |B| * 0 = 0.