You are given two vectors A = -3i + 6j and B = 7i + 2j. Let Counterclockwise angles be positive (a) What angle does A make with the +x-axis?

Verified step by step guidance

1. Identify the components of vector A, which are Ax = -3 (along the i-axis) and Ay = 6 (along the j-axis).
2. Calculate the angle \( \theta \) that vector A makes with the positive x-axis using the tangent function: \( \tan(\theta) = \frac{Ay}{Ax} \).
3. Substitute the values of Ax and Ay into the tangent function: \( \tan(\theta) = \frac{6}{-3} \).
4. Determine the angle \( \theta \) by taking the arctangent (inverse tangent) of the result from the previous step: \( \theta = \tan^{-1}\left(\frac{6}{-3}\right) \).
5. Since the angle calculated will be in the fourth quadrant (because Ax is negative and Ay is positive), adjust the angle to find the counterclockwise angle from the positive x-axis.

Key Concepts

Vector Representation

Vectors are quantities that have both magnitude and direction, represented in a coordinate system. In this case, vectors A and B are expressed in terms of their components along the x-axis (i) and y-axis (j). Understanding how to interpret these components is essential for calculating angles and performing vector operations.

Angle with the x-axis

The angle a vector makes with the positive x-axis can be determined using the arctangent function. Specifically, for a vector A with components (Ax, Ay), the angle θ is given by θ = arctan(Ay/Ax). This angle is measured counterclockwise from the positive x-axis, which is crucial for determining the orientation of the vector in the Cartesian plane.

Trigonometric Functions

Trigonometric functions, particularly sine, cosine, and tangent, are fundamental in relating angles to the ratios of the sides of right triangles. In the context of vectors, these functions help in calculating angles and understanding the geometric relationships between vectors and their components. They are essential for solving problems involving angles and directions in physics.