Consider vector A⃗ with components Ax and Ay. Derive an expression, in terms of Ax and Ay, for the components of B⃗ (Bx and By) if B⃗ satisfies the vector equation A⃗ + 4B⃗ = 3A⃗ . Express your answer in terms of Ax and Ay separated by a comma.

Glass is transparent to visibile light under normal conditions; however, at extremely high intensities, glass will absorb most of the light incident upon it. this works through a process known as multiphoton absorption. in this process, several photons are absorbed at the same time. if very intense light whose photons carry 2ev of energy is shined onto a material with a band gap of 4ev, that light can be absorbed through two-photon absorption, because two photons have the right amount of energy to bridge the band gap. what is the minimum number of photons of 800-nm light that are needed to equal or exceed the band gap of fused silica glass

J. j. thomson’s experiment disproved the theory that an atom

The scalar triple product computes the magnitude m of the moment of a force vector f about a specified line. it is m = (r × f) · n, where r is the position vector from the line to the point of application of the force and n is a unit vector in the direction of the line. use matlab to compute the magnitude m for the case where f = [12, −5, 4] n, r = [−3, 5, 2] m, and n = [6, 5, −7].