A protester carries his sign of protest, starting from the origin of a 3D-Cartesian coordinate system,with the xy plane horizontal. He moves 40 m in the negative direction of the x-axis, then 20 m alonga perpendicular path to his left, and then 25 m up a water tower.(a) What does the scalar product of a vector, with itself, give? What about the vector product of avector with itself? Explain using no equations. (1)(b) In unit-vector notation, what is the displacement of the sign from start to end? (2)(c) The sign then falls to the foot of the tower. What is the magnitude of the displacement of thesign from the start to this new end? (3)(d) Verify the anti-commutative property of cross multiplication with the 1st and 2nd displacementvectors you got from (b) and (c), respectively.

(a) The scalar product of a vector with itself gives the square of the magnitude of the vector. It represents the dot product of the vector with itself, resulting in a scalar value. The vector product of a vector with itself, on the other hand, is not defined. The vector product, also known as the cross product, is a binary operation between two vectors that results in a new vector perpendicular to both input vectors. Since the cross product requires two distinct vectors, it cannot be applied to a vector with itself.

(b) In unit-vector notation, the displacement of the sign from start to end can be represented as follows: -40i + 20j + 25k Here, i, j, and k represent the unit vectors in the x, y, and z directions respectively. The coefficients (-40, 20, 25) indicate the magnitudes and directions of the displacements along each axis.

(c) The magnitude of the displacement of the sign from the start to the new end, after it falls to the foot of the tower, can be calculated using the Pythagorean theorem. The displacement along the x-axis is -40 m, the displacement along the y-axis is 20 m, and the displacement along the z-axis is 25 m. Therefore, the magnitude of the displacement is given by: √((-40)^2 + 20^2 + 25^2)

(d) To verify the anti-commutative property of cross multiplication, we need to calculate the cross product of the 1st and 2nd displacement vectors obtained from (b) and (c) respectively. Let's denote the 1st displacement vector as A and the 2nd displacement vector as B. Then, the cross product of A and B is given by: A x B = (-40i + 20j + 25k) x (A_xi + A_yj + A_zk)

We can also calculate the cross product of B and A: B x A = (A_xi + A_yj + A_zk) x (-40i + 20j + 25k)

If the anti-commutative property holds, then A x B should be equal to -1 times B x A. We can verify this by calculating both cross products and checking if they are equal.