FIGURE EX3.19 shows vectors A and B. What is C = A + B? Write your answer in component form using unit vectors

Verified step by step guidance

1. Identify the components of vectors A and B from the figure. Typically, these vectors will be given in terms of their x (horizontal) and y (vertical) components, such as A = A_x \hat{i} + A_y \hat{j} and B = B_x \hat{i} + B_y \hat{j}, where \hat{i} and \hat{j} are the unit vectors in the x and y directions respectively.
2. Add the corresponding components of vectors A and B to find the x and y components of vector C. Use the formula C_x = A_x + B_x for the x-component and C_y = A_y + B_y for the y-component.
3. Combine the components you found in the previous step to write vector C in component form. The result will be C = C_x \hat{i} + C_y \hat{j}.
4. Check the direction and magnitude of vector C to ensure it makes sense given the directions and magnitudes of vectors A and B. The direction of C should be somewhere between the directions of A and B unless one vector is much larger than the other.
5. If necessary, draw the resultant vector C on the same coordinate system as vectors A and B to visually verify the addition and direction.

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. This is done by adding the corresponding components of the vectors. For example, if vector A has components (Ax, Ay) and vector B has components (Bx, By), the resultant vector C can be expressed as C = (Ax + Bx, Ay + By).

Unit Vectors

Unit vectors are vectors that have a magnitude of one and are used to indicate direction. In a Cartesian coordinate system, the standard unit vectors are i (for the x-direction) and j (for the y-direction). Any vector can be expressed in terms of unit vectors, such as A = Ax i + Ay j, which simplifies the representation of vectors in component form.

Component Form

Component form is a way of expressing vectors in terms of their horizontal and vertical components. For a vector C resulting from the addition of vectors A and B, the component form is written as C = Cx i + Cy j, where Cx and Cy are the sums of the respective components of A and B. This format is essential for performing calculations and visualizing vector operations.