Find the angles that vector D = (2.30
i − 4.50
j + 1.10
k) m makes with the x
-, y
-, and z
- axes.

Find the angle that the vector D makes with the y
-axis.

To find the angle that the vector \( \mathbf{D} = (2.30 \hat{i} - 4.50 \hat{j} + 1.10 \hat{k}) \, \text{m} \) makes with the \( y \)-axis, we use the formula for the angle between a vector and an axis:

\[
\cos \theta_y = \frac{D_y}{|\mathbf{D}|}
\]

where:
- \( D_y = -4.50 \, \text{m} \) is the \( y \)-component of the vector,
- \( |\mathbf{D}| \) is the magnitude of the vector \( \mathbf{D} \).

First, let's calculate the magnitude of \( \mathbf{D} \):

\[
|\mathbf{D}| = \sqrt{D_x^2 + D_y^2 + D_z^2}
\]

Substituting the components \( D_x = 2.30 \, \text{m} \), \( D_y = -4.50 \, \text{m} \), and \( D_z = 1.10 \, \text{m} \):

\[
|\mathbf{D}| = \sqrt{(2.30)^2 + (-4.50)^2 + (1.10)^2}
\]

Let's calculate the magnitude first.The magnitude of the vector \( \mathbf{D} \) is approximately \( 5.17 \, \text{m} \).

Now, using the formula \( \cos \theta_y = \frac{D_y}{|\mathbf{D}|} \), we can calculate the angle \( \theta_y \):

\[
\cos \theta_y = \frac{-4.50}{5.17}
\]

Let's compute \( \theta_y \).It seems like I can’t do more advanced data analysis right now. Please try again later. If you want, I can guide you through the calculation manually.

Would you like to proceed with a step-by-step explanation or ask about another problem?

Here are some related questions you might want to explore:
1. What is the angle between the vector and the \( x \)-axis?
2. How do you compute the angle between any vector and a coordinate axis?
3. What is the significance of the dot product in angle calculations for vectors?
4. How can vectors be used in real-life applications like physics or engineering?
5. What happens to the angle calculation if the vector components are negative?

**Tip**: Always ensure you are using the right units when performing vector calculations to avoid mistakes.

Find the angles that vector D = (2.30 i − 4.50 j + 1.10 k) m makes with the x-, y-, and z- axes. Find the angle that the vector D makes with the y-axis.

Learn how to calculate the angle between a vector and the y-axis using vector geometry and trigonometry. This problem demonstrates how to find the angle with the y-axis for vector D = (2.30 i − 4.50 j + 1.10 k) m. Key concepts include the cosine rule for vectors and calculating the magnitude of a vector. Suitable for Grades 11-12.

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