Use components to determine the magnitude and direction of G = E＋F.

Verified step by step guidance

1. Break down the given vectors E and F into their components. For each vector, use the equations: $x=magnitude\cdot \cos \left(\theta \right)$ for the x-component and $y=magnitud\cdot \sin \left(\theta \right)$ for the y-component, where $\theta$ is the angle of the vector with respect to the positive x-axis.
2. Add the x-components of vectors E and F to find the x-component of the resultant vector G: $G_x=E_x+F_x$.
3. Add the y-components of vectors E and F to find the y-component of the resultant vector G: $G_y=E_y+F_y$.
4. Determine the magnitude of the resultant vector G using the Pythagorean theorem: $G=\sqrt{{G_x}^2+{G_y}^2}$.
5. Find the direction (angle) of the resultant vector G with respect to the positive x-axis using the inverse tangent function: $\theta =\tan ⁻1\left(\frac{G_y}{G_x}\right)$. Ensure the angle is expressed in the correct quadrant based on the signs of $G_x$ and $G_y$.

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. This involves adding the corresponding components of the vectors, typically represented in a Cartesian coordinate system. The magnitude and direction of the resultant vector can be found using the Pythagorean theorem and trigonometric functions, which are essential for understanding how forces interact in physics.

Components of a Vector

Vectors can be broken down into their components along the axes of a coordinate system, usually the x and y axes. Each vector can be expressed as a sum of its horizontal (x) and vertical (y) components, allowing for easier calculations in physics. Understanding how to resolve vectors into components is crucial for accurately determining the resultant vector's magnitude and direction.

Resultant Vector

The resultant vector is the single vector that has the same effect as two or more vectors acting together. It is obtained by vector addition and represents both the total magnitude and direction of the combined vectors. In the context of forces, the resultant vector indicates the net force acting on an object, which is vital for analyzing motion and equilibrium in physics.