Calculating the Angle Between a Vector and the Positive Y-Axis

Understanding the angle a vector makes with the positive y-axis is an essential skill in vector mathematics. This article will walk through the steps and provide the necessary calculations using the vector a 4i - j 3k as an example.

1. Introduction to Vector Angles and the Y-Axis

The angle between a vector A and the positive y-axis can be determined using the dot product and the magnitudes of the vectors. The formula to find the angle theta between two vectors A and B is given by:

cos(theta) (A · B) / (|A| |B|)

The dot product of two vectors A and B is calculated as:

A · B |A||B|cos(theta)

2. Using the Given Vector a 4i - j 3k

Let's calculate the angle between the vector A 4i - j 3k and the positive y-axis, which is represented by the vector B j 0 i 1 0.

2.1 Dot Product Calculation

The dot product of vectors A and B is given by:

A · B (4i - j 3k) · (0i 1j 0k) 4*0 -1*1 3*0 -1

2.2 Magnitude Calculations

The magnitude of vector A is:

|A| sqrt{4^2 (-1)^2 3^2} sqrt{16 1 9} sqrt{26}

The magnitude of vector B is:

|B| sqrt{0^2 1^2 0^2} 1

2.3 Substitution and Calculation

Substitute the values into the cosine formula for theta:

cos(theta) (A · B) / (|A| |B|) -1 / (sqrt{26} * 1) -1 / sqrt{26}

The angle theta can be found using the arccos function:

theta arccos(-1 / sqrt{26}) 103.6 degrees

3. Conclusion

Thus, the vector A 4i - j 3k makes an angle of approximately 103.6 degrees with the positive y-axis.

4. Relevance and Applications

Understanding vector angles is crucial in physics and engineering, particularly in 3D space. For example, the concept can be applied to analyze forces, torque, and rotational dynamics in mechanical systems. Proficiency in these calculations can aid in designing and optimizing complex systems.

5. Further Reading and Resources

For those interested in delving deeper into vector mathematics, there are numerous resources available online and in textbooks. Keywords and concepts to explore further include:

Vector projection Vector magnification and normalization Vector resolution in different axes Applications of vector angles in real-world scenarios