Final answer:
In answering the vector equations, vector addition and subtraction principles are applied, determining R by manipulating vector D and F and using the head-to-tail method to determine the resultant vector. Components of vectors are summed in accordance with vector addition methods to find magnitudes and directions of resultant vectors.
Step-by-step explanation:
To solve the vector equations given, we'll need to apply vector addition and subtraction principles. For equation (a) Đ + R = F, R can be found by subtracting the vector Đ from vector F. This can be written as R = F - Đ, which means the operation involves inverting the direction of vector Đ and then using the head-to-tail method to find the resultant vector R.
For equation (b) Ĉ - 2Đ + 5Ŕ = 3F, we'll first multiply each vector by its respective scalar, then add or subtract the vectors accordingly. The resultant vector can again be found using the head-to-tail method. It's important to determine the components (such as Rx and Ry) to find the magnitude and direction of the resultant. The analytical method of vector addition will help us determine these components.
By combining vectors, for example in the equation Ŕ = Ē + B, the final resultant vector R is obtained by adding the corresponding scalar components. Applying Equation 2.24, the scalar components of the resultant vector R are RxĠ + RyĴ + R₂Â, which are the sums of corresponding scalar components of vectors A and B.
Using this method, we can find the magnitude and direction of any resultant vector. An example is given with components Cx = −2/3, Cy = −4/3, and C₂ = 7/3, which can be substituted into the relevant equation to calculate the magnitude of vector C.