https://file.aiquickdraw.com/mathbot/file/1726817158897a0ti3xjq.jpg

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.

Learn how to find the Cartesian representation of a vector in 3D space. In this example, we decompose a 4 kN vector into its x, y, and z components using trigonometry. This detailed explanation covers key vector decomposition techniques and is suitable for students at the high school or early college level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation