Give a mathematical problems for vector quantity
1. Given vectors A = 3i + 4j - 2k and B = 2i - j + 5k, calculate the dot product of A and B.
2. Two forces acting on a body are given by vectors F1 = 2i + 3j - 4k and F2 = -4i + 2j + k. Find the resultant force acting on the body by adding these two vectors.
3. A vector C = 5i - 2j + 3k is perpendicular to vector D = 2i + 4j + 3k. Calculate the scalar product of C and D.
4. A particle moves in a straight line with velocity vector v = (3t)i + (4 - t²)j + (2 + 2t)k, where t is time. Determine the acceleration of the particle.
5. A boat is moving at a velocity vector v = 6i + 2j - 3k with respect to the water. The water is flowing with a velocity vector w = 3i - 2j + 4k with respect to the ground. Find the boat's velocity vector with respect to the ground.
6. The position vector of a particle is given by r(t) = (2t²)i + (4t - 3)j + (t³ - 2)k, where t is time. Determine the velocity and acceleration of the particle.
7. Three parallel forces are acting on a body in the same direction with magnitudes 5N, 7N, and 10N. If the distance between the forces is 2m, what is the resultant force on the body?