Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.


<IMAGE>


a + (b + c)

Verified step by step guidance

1. Start by sketching vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) on a coordinate plane or a piece of paper.
2. To find \( \mathbf{b} + \mathbf{c} \), place the initial point of vector \( \mathbf{c} \) at the terminal point of vector \( \mathbf{b} \).
3. Draw the resultant vector from the initial point of \( \mathbf{b} \) to the terminal point of \( \mathbf{c} \). This is the vector \( \mathbf{b} + \mathbf{c} \).
4. Now, place the initial point of vector \( \mathbf{a} \) at the initial point of the resultant vector \( \mathbf{b} + \mathbf{c} \).
5. Use the parallelogram rule: draw a parallelogram with \( \mathbf{a} \) and \( \mathbf{b} + \mathbf{c} \) as adjacent sides. The diagonal of the parallelogram from the common initial point is the resultant vector \( \mathbf{a} + (\mathbf{b} + \mathbf{c}) \).

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. This can be done graphically by placing the tail of one vector at the head of another, or by using the parallelogram rule, where two vectors are represented as adjacent sides of a parallelogram, and the diagonal represents the resultant.

Parallelogram Rule

The parallelogram rule is a method for finding the resultant of two vectors. By drawing a parallelogram where the two vectors are adjacent sides, the diagonal from the common initial point to the opposite corner represents the resultant vector. This rule is particularly useful for visualizing vector addition in two dimensions.

Associative Property of Vector Addition

The associative property of vector addition states that the way in which vectors are grouped does not affect the resultant. For example, (a + b) + c is the same as a + (b + c). This property allows for flexibility in calculations and simplifies the process of adding multiple vectors.