To find the Cartesian representation of the vector $Q$, we need to decompose it into its components along the $x$, $y$, and $z$ axes. Based on the diagram, vector $Q$ has a magnitude of $4 \, \text{kN}$, and the angles provided are $60^\circ$ from the $z$-axis and $60^\circ$ from the $y$-axis.

Let’s define the Cartesian components of $Q$ as:

$Q_x = Q \cdot \cos(\theta_x)$ $Q_y = Q \cdot \cos(\theta_y)$ $Q_z = Q \cdot \cos(\theta_z)$

Here, $Q = 4 \, \text{kN}$, and the angles for the projection in the $x$, $y$, and $z$ axes are as follows:

• $\theta_y = 60^\circ$ (angle between $Q$ and the $y$-axis)
• $\theta_z = 60^\circ$ (angle between $Q$ and the $z$-axis)
• $\theta_x = 90^\circ - 45^\circ = 45^\circ$ (since $45^\circ$ is the angle between the projection in the $x$-$y$ plane and the $x$-axis)

Now let’s compute the components.

$Q_x = 4 \cdot \cos(45^\circ)$ $Q_y = 4 \cdot \cos(60^\circ)$ $Q_z = 4 \cdot \cos(60^\circ)$

Let me calculate these values for you.The Cartesian representation of the vector $Q$ is approximately:

$Q_x = 2.83 \, \text{kN}$ $Q_y = 2.00 \, \text{kN}$ $Q_z = 2.00 \, \text{kN}$

Thus, the vector $Q$ in Cartesian form is:

$Q = (2.83 \, \hat{i} + 2.00 \, \hat{j} + 2.00 \, \hat{k}) \, \text{kN}$

Would you like further details or have any questions?

Here are some related questions to expand on this concept:

1. How would you find the magnitude of a vector from its Cartesian components?
2. Can you explain how to determine the direction cosines of a vector?
3. How would the representation change if the angle with one of the axes were different?
4. How do these vectors combine when multiple forces act on a body?
5. How can the dot product be used to find angles between two vectors?

Tip: When breaking down vectors in 3D, it's useful to verify each component's magnitude to ensure correct orientation.