Let A = 4i - 2j, B = -3i + 5j, and C = A + B. (a) Write vector C in component form.

Verified step by step guidance

1. Identify the components of vectors A and B. Vector A has components (4, -2) and vector B has components (-3, 5).
2. To find vector C, which is the sum of vectors A and B, add the corresponding components of A and B. Add the i-components together and the j-components together.
3. Add the i-components of A and B: 4i + (-3i).
4. Add the j-components of A and B: -2j + 5j.
5. Combine the results from the previous steps to write vector C in component form.

Key Concepts

Vector Addition

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

Component Form of a Vector

The component form of a vector expresses it in terms of its horizontal and vertical components, typically represented as A = xi + yj, where x is the horizontal component and y is the vertical component. This form allows for easier calculations, especially in vector addition and subtraction.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction. The standard unit vectors in two-dimensional space are i (representing the x-direction) and j (representing the y-direction). They are essential for expressing vectors in component form and simplifying vector operations.