Calculate, correct to one decimal place, the angle between 5i + 12j and -2i + 3j.

To find the angle between two vectors, we can use the dot product formula: cos θ = (A dot B) / (|A| |B|) where A and B are the two vectors, A dot B is their dot product, and |A| and |B| are their magnitudes. Let's apply this formula to the given vectors: A = 5i + 12j B = -2i + 3j A dot B = (5*-2) + (12*3) = -10 + 36 = 26 |A| = sqrt(5^2 + 12^2) = 13 |B| = sqrt((-2)^2 + 3^2) = sqrt(13) cos θ = (26) / (13 * sqrt(13)) Using a calculator, we can find that cos θ ≈ 0.801 To find the angle θ itself, we can take the inverse cosine (cos^-1) of this value: θ ≈ cos^-1(0.801) ≈ 37.8° However, this is only half of the actual angle between the two vectors. To get the full angle, we need to take into account the direction of the vectors. If we draw the two vectors on a coordinate plane, we can see that they are in different quadrants: A is in the second quadrant and B is in the fourth quadrant. The angle between them will be the difference between their angles relative to the positive x-axis. Using the inverse tangent (tan^-1) function, we can find the angles of each vector: Angle of A = tan^-1(12/5) ≈ 67.4° Angle of B = tan^-1(-3/2) ≈ -56.3° (Note that we use a negative angle for B because it is in the fourth quadrant, where angles are negative.) The angle between the vectors will be the difference between these two angles: θ = |67.4° - (-56.3°)| = 123.7° However, this is the full angle between the vectors, and the question only asks for the acute angle (the smaller of the two possible angles). To find the acute angle, we subtract the full angle from 180°: Acute angle = 180° - 123.7° ≈ 56.3° Therefore, the answer is option B: 56.3°.