Question

Vector A has magnitude 4.0 units and vector B has magnitude 6.0 units. The angle between A and B is 60.0°. What is the magnitude of 2A+3B ?

Answer 1

Let's direct the x-axis of Cartesian coordinate along vector $\mathbf{A}$. Then the coordinates of vector $\mathbf{A}$ will be:

$$\mathbf{A} = 4(1,0) = (4,0)$$

The coondinates of vector $\mathbf{B}$ are;

$$\mathbf{B} = 6(\cos60\degree,\sin60\degree) = (3\sqrt{3},3)$$

Then the combitanion will have the following coordinates:

$$2\mathbf{A} + 3\mathbf{B} = 2(4,0) + 3(3\sqrt{3},3) = (8+9\sqrt3 , 9)$$

The magnitude of this vector is:

$$|2\mathbf{A} + 3\mathbf{B}| = \sqrt{(8+9\sqrt{3})^2 + 9^2} \approx 25.2$$

Answer. 25.2.