The stock mounted on the lathe is subjected to a force of 60 N. Determine the coordinate direction angle β and express the force as a Cartesian vector.

#### Solution:

### To solve this question, we must remember the following equation.

### \text{cos}^2\alpha +\text{cos}^2\beta +\text{cos}^2\gamma =1

### We will simply substitute the values shown in the diagram above.

### \alpha =60^0

### \gamma =45^0

### Thus, we can write:

### 1=\text{cos}^{2}(60^0) +\text{cos}^2(\beta) +\text{cos}^{2}(45^0)

### Isolating for \text{cos}\beta gives us:

### \text{cos}\beta=\pm 0.5

### Solving for \beta gives us:

### \beta =60^0, 120^0

### Using \beta =120^0, we can write the force as a Cartesian vector.

### F=60\,N(\text{cos}60^0\vec i+\text{cos}120^0\vec j+\text{cos}45^0\vec k)

##### (we simply expand the brackets by multiplying each value in the brackets by 60 N)

how do I know to choose 120 not 60, I find this very difficult to do to related questions

Hi,

Thanks for visiting our site. The easiest answer I can give you is to look at the diagram. In every question, the diagram will show where the force is heading so it’s easy to see what angle to use. For this question, we see that the force is heading into the negative y-axis and positive x-axis. So the angle that the force makes with the y-axis must be greater than 90°. I hope that makes sense. If you need further clarifications, please let us know.

Best of luck with your studies! 🙂