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?

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1. To find the angle that vector A makes with the +x-axis, we need to use the tangent function, which relates the components of the vector to the angle. The formula is: \( \tan(\theta) = \frac{A_y}{A_x} \), where \( A_x \) and \( A_y \) are the x and y components of vector A, respectively.
2. For vector A = -3i + 6j, the x-component \( A_x = -3 \) and the y-component \( A_y = 6 \). Substitute these values into the tangent formula: \( \tan(\theta) = \frac{6}{-3} \).
3. Calculate the value of \( \tan(\theta) \): \( \tan(\theta) = -2 \).
4. To find the angle \( \theta \), take the arctangent (inverse tangent) of -2: \( \theta = \tan^{-1}(-2) \).
5. Since the tangent function is negative and we are considering counterclockwise angles as positive, the angle \( \theta \) will be in the second quadrant. Adjust the angle accordingly to ensure it is measured counterclockwise from the +x-axis.

Key Concepts

Vector Components

Vectors are mathematical entities with both magnitude and direction, represented in terms of components along the coordinate axes. For a vector A = -3i + 6j, -3 and 6 are the components along the x and y axes, respectively. Understanding these components is crucial for calculating angles and magnitudes.

Dot Product and Angle Calculation

The dot product of two vectors is a scalar value that can be used to find the angle between them. For a vector A with components (Ax, Ay), the angle θ with the x-axis can be found using the formula cos(θ) = Ax/|A|, where |A| is the magnitude of A. This relationship helps determine the angle A makes with the x-axis.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector A = -3i + 6j, the magnitude |A| is √((-3)^2 + 6^2) = √(9 + 36) = √45. Knowing the magnitude is essential for normalizing the vector and calculating angles.