Vector A has a magnitude of 5.00 units, vector has a magnitude of 5.00 units, and the dot product A·B has a value of 16. What is the angle between the directions of A and ?
A•B= |A||B| cosα
cosα =A•B/ |A||B|=
=16/5•5=0.64
α=cos⁻¹α=50.2°
A person walks in the following pattern: 2.8 km north, then 2.7 km west, and finally 6.9 km south. Construct the vector diagram that represents this motion and from it judge how far and in what direction a bird would fly in a straight line from the same starting point to the same final point. (Choose east as the +x direction.)
To find the angle between the directions of vectors A and B, we need to use the dot product formula:
A · B = |A| |B| cos θ
Given that the magnitude of vector A and B is 5.00 units and the dot product A · B is 16:
16 = 5.00 * 5.00 * cos θ
Let's solve for cos θ first:
cos θ = 16 / (5.00 * 5.00)
cos θ = 0.64
Now, to find the angle θ, we need to take the inverse cosine (cos^-1) of 0.64:
θ = cos^-1(0.64)
Using a calculator or a mathematical software, you can find that the angle is approximately 49.4 degrees.
So, the angle between the directions of vectors A and B is approximately 49.4 degrees.