In Exercises 1-26, assume that the coordinates of the points...

In Exercises $1-26,$ assume that the coordinates of the points $P$, $Q, R, S,$ and $O$ are as follows: $$P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)$$ For each exercise, draw the indicated vector (using graph paper and compute its magnitude. In Exercises $7-20$, compute the sums using the definition given on page $698 .$ In Exercises $21-26,$ use the parallelogram law to compute the sums. $$\overrightarrow{O P}+\overrightarrow{O Q}$$

In Exercises $1-26,$ assume that the coordinates of the points $P$, $Q, R, S,$ and $O$ are as follows:
$$P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)$$
For each exercise, draw the indicated vector (using graph paper and compute its magnitude. In Exercises $7-20$, compute the sums using the definition given on page $698 .$ In Exercises $21-26,$ use the parallelogram law to compute the sums.
$$\overrightarrow{O P}+\overrightarrow{O Q}$$

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Key Concepts

Vector Representation

A vector is an object that has both a magnitude and a direction. In a coordinate plane, vectors can be represented by their components along the x and y axes, which makes it possible to perform operations like addition by handling each component independently.

Vector Addition

Vector addition involves combining two or more vectors to produce a new vector. This is typically done by adding the corresponding components of the vectors. The resultant vector embodies the cumulative effect of the individual vectors, both in terms of magnitude and direction.

Magnitude of a Vector

The magnitude of a vector is its length, which can be computed using the Pythagorean theorem when the vector is represented in terms of its x and y components. Calculating the magnitude provides a quantitative measure of the vector's size.

Parallelogram Law

The parallelogram law is a graphical method for vector addition. When two vectors are positioned so that they share an initial point, they can be arranged to form two adjacent sides of a parallelogram. The diagonal of the parallelogram, starting from the common initial point, represents the sum of the two vectors.

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