Question

1 Vector Operations H4 C4 HS H3 C3 C2 H6 H2 C1 H1 The positions (r,y, z) of each of the atoms in a given benzene molecule are known at a given moment of time, t, from a molecular dynamics simulation. If we are explicitly given the position vectors of the C1, C2 and H1 atoms: rH1 3.348ex 13.706ey + 27.438e find the following quantitities: 1. Vector Addition a. Determine the vector pointing from C1 to C2 b. Determine the vector pointing from Cl to H1 2. Norm a. Determine the C1 -C2 bond length. b. Determine the C1-H1 bond length. 3. Dot Product & Angle Between Two Vectors. a. Determine the C2 C1-H1 angle in both radians and degrees.

Answer 1

The key concept we will use is law of parallelogram for vector addition.

"Given r_c_2 = 4.603 e_x + 11.977 e_y + 27.383 e_z r_c_1 = 3.673 e_x + 12.860 e_y + 26.859 e_z r_H_1 = 9.348 e_x + 13.706 e_y + 27.438 e_z a The vector from c_1 to c_2 is say c_1c_2 Then, by vector addition, rc_1 + c_1c_2 = rc_2 therefore c_1c_2 = rc_2-rc_1 = (4.603-3.673)e_x + (11.977-12.860)e_y + (27.383-26.859)e_z therefore c_1c_2 = 0.93e_x-0.883e_y + 0.524e_z b If vector from c_1 to H_1 is c_1 H_1 then by vector addition rc_1 + c_1H_1 = rH_1 therefore c_1H_1 = rH_1-rc_1 = (3.348-3.673)e_x + (13.706-12.860)e_y + (27.438-26.859)e_z therefore c_1H_1 = -0.325e_x + 0.846e_y + 0.579e_z \r\n a length of c_1-c_2 =