Find the magnitude and direction of the vector represented by the following pairs of components: A_x = −8.60 cm, A_y = 5.20 cm

Verified step by step guidance

1. Step 1: Understand that the vector components Ax and Ay represent the horizontal and vertical components of the vector, respectively. The magnitude of the vector can be found using the Pythagorean theorem.
2. Step 2: Use the formula for the magnitude of a vector: $\sqrt{A^2+B^2}$, where A and B are the components Ax and Ay. Substitute Ax = -8.60 cm and Ay = 5.20 cm into the formula.
3. Step 3: Calculate the magnitude of the vector using the formula: $\sqrt{{-}^{8.60}+{5.20}^2}$ cm.
4. Step 4: Determine the direction of the vector using the tangent function. The direction angle θ can be found using the formula: $\theta =\tan -1\left(\frac{\mathrm{Ay}}{\mathrm{Ax}}\right)$. Substitute Ax = -8.60 cm and Ay = 5.20 cm into the formula.
5. Step 5: Calculate the direction angle θ using the formula: $\theta =\tan -1\left(\frac{5.20}{-}\right)$. Consider the signs of Ax and Ay to determine the correct quadrant for the angle.

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector with components Ax and Ay, the magnitude is given by the formula √(Ax² + Ay²). This provides a scalar quantity representing the size of the vector irrespective of its direction.

Vector Direction

The direction of a vector is determined by the angle it makes with a reference axis, typically the positive x-axis. This angle, θ, can be found using trigonometry, specifically the tangent function: θ = arctan(Ay/Ax). The angle provides insight into the vector's orientation in a coordinate system.

Coordinate System

Understanding the coordinate system is crucial for interpreting vector components. In a Cartesian coordinate system, vectors are expressed in terms of their horizontal (x-axis) and vertical (y-axis) components. This system allows for the decomposition and analysis of vectors in two dimensions, facilitating calculations of magnitude and direction.