Express the force as a Cartesian vector.

Steps to Solve

1. Identify the Components of the Triangle

From the provided diagram, we can see that the force vector $F = 50$ lb forms a triangle with the angles and sides depicted. The horizontal component in the x-y plane is represented by a right triangle where one leg is adjacent to the angle of $45^\circ$.

2. Determine the Projection on the x-y plane

Using the angles and the lengths provided:

• The length in the x-direction is represented as $x = 5$ (since the right triangle indicates a base of 5).
• The length in the y-direction is represented as $y = 4$.

This means the force vector’s projection can be calculated using the proportions of the triangle.

3. Calculate the Magnitude of the Components

The magnitude of the components in the x and y directions can be calculated with the following equations:

$$ F_x = F \cdot \frac{x}{\sqrt{x^2 + y^2}} $$

$$ F_y = F \cdot \frac{y}{\sqrt{x^2 + y^2}} $$

For the z-direction, the angle with respect to the z-axis is also $45^\circ$, so we have:

$$ F_z = F \cdot \frac{\text{height}}{F} $$

Since the force is acting upwards, and height can be derived from the trigonometric function of the triangle.

4. Substituting Values

Now we need to substitute:

The total length of the vector $F = 50$ lb, then use:

$$ F_x = 50 \cdot \frac{5}{\sqrt{5^2 + 4^2}} $$

$$ F_y = 50 \cdot \frac{4}{\sqrt{5^2 + 4^2}} $$

Since the z component is determined by single 45° projection, we can have:

$$ F_z = 50 \cdot \frac{height}{50} $$ (here height is equivalent to the projection direction)

5. Final Component Calculations

Calculate the values for all components:

• Compute for $F_x$ and $F_y$ using the triangle’s dimensions.
• For $F_z$ it can be approximately derived, hence $F_z = F$ maintains this direct 45° direction in upwards.

Thus the components can be derived to form the final vector expression in Cartesian coordinates:

$$ \vec{F} = \langle F_x, F_y, F_z \rangle $$