Question

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.

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a + (b + c)

Accepted Answer

Begin by sketching each of the vectors $$ \mathbf{a} $$, $$ \mathbf{b} $$, and $$ \mathbf{c} $$ separately, paying attention to their directions and magnitudes as given or implied in the problem. Next, focus on the expression inside the parentheses: $$ \mathbf{b} + \mathbf{c} . To $$add these two vectors, place the initial point of $$ \mathbf{c}  at $$the terminal point of $$ \mathbf{b} . $$Use the parallelogram rule or the tip-to-tail method to find the resultant vector $$ \mathbf{b} + \mathbf{c} . $$This resultant vector starts at the initial point of $$ \mathbf{b} $$ and ends at the terminal point of $$ \mathbf{c} $$ after placement. Now, to find $$ \mathbf{a} + (\mathbf{b} + \mathbf{c}) $$, place the initial point of the resultant vector $$ \mathbf{b} + \mathbf{c}  at $$the terminal point of $$ \mathbf{a} . $$Finally, draw the resultant vector starting from the initial point of $$ \mathbf{a}  to $$the terminal point of $$ \mathbf{b} + \mathbf{c} $$ after placement. This vector represents $$ \mathbf{a} + (\mathbf{b} + \mathbf{c}) .$$

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Key Concepts

Vector Addition

Parallelogram Rule

Associative Property of Vector Addition