Determine the angle between the vectors u and v. u=5i+2j,v=3i−7j

Answer 1

The dot product of the two vectors u and v is given by

u_i*v_{i +} u_j*v_{j = 5*3 + 2*(-7) = 15 - 14 = 1}

_{The dot product can also be determined by}

_{|u| * |v| * cos θ = 1}

|u| = square root [(5)² + (2)²]= square root (29)

|v| = square root [(3)² + (-7)²] = square root (58)

So substituting for |u| and |v| and solving for θ yields

square root (29) * square root (58) * cos θ = 1

cos θ = 1/square root (29*58)

θ = cos^-1 (1/square root (1682)

θ = 88.6 deg

Answer 2

|u| = sqrt[(5)² + (2)²] = sqrt(29)

|v| = sqrt[(3)² + (-7)²] = sqrt(55)

cos(θ) = u.v / |u||v| = (5)(3) + (2)(-7) / [sqrt(29).sqrt(55)] = 1 / sqrt[(29)(55)] = 0.025

θ = cos^-1(0.025) = 88.6 degrees