5-2-vector-addition-and-subtraction-analytical-methods_summary

This section is about Vector Addition and Subtraction: Analytical Methods. The goal is to understand the analytical method of adding and subtracting vectors, and use it to solve problems. This method uses the trigonometric relationships of sine, cosine, and tangent, and the Pythagorean theorem to find the components of a vector. The standard covered includes expressing and interpreting relationships symbolically to make predictions and solve problems mathematically, as well as identifying and describing motion relative to different frames of reference. Key terms include Components of Vectors, which are pieces of a two-dimensional vector that point in the x- or y-direction, and can be used to express any 2-d vector as a sum of its x and y components. To find the components (A_x and A_y) of a vector with magnitude A and given direction, the following relationships can be used: A_x = A*cos(θ) and A_y = A*sin(θ), where θ is the angle of the resultant vector with respect to the x-axis. The magnitude and direction of the resultant vector can be found using the Pythagorean theorem: R = sqrt(A_x^2 + A_y^2) and θ = tan^-1(A_y/A_x) To add or subtract two vectors, the components of the resultant vector can be found using the same relationships (A_x = A_1*cos(θ_1) + A_2*cos(θ_2) and A_y = A_1*sin(θ_1) + A_2*sin(θ_2)), where A_1 and A_2 are the magnitudes of the two vectors being added, and θ_1 and θ_2 are their respective directions. Misconceptions to be aware of include confusing the relationship between the magnitude and direction of a vector with the addition of magnitudes of vectors, and the assumption that the graphical method is more accurate than the analytical method. The graphical method is more accurate than the analytical method due to the precision of the drawing, rather than the reverse.