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. $$\overrightarrow{O P}+\overrightarrow{Q R}$$

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.
$$\overrightarrow{O P}+\overrightarrow{Q R}$$

Key Concepts

Parallelogram Law

The parallelogram law is a geometrical method of vector addition. When two vectors are positioned so that they start at the same point, a parallelogram can be constructed with these vectors as adjacent sides. The diagonal of the parallelogram that starts from the same point represents the resultant vector. This visual method complements the component-wise addition technique and is especially useful for understanding the geometric interpretation of vector addition.

Vector Magnitude

The magnitude of a vector is the distance from its tail to its head, effectively measuring its length. It is calculated using the Pythagorean theorem based on the vector's components. Understanding magnitude is crucial for quantifying the size of the vector and comparing different vectors.

Vector Representation in the Coordinate Plane

Vectors are represented in the coordinate plane as directed line segments with both magnitude and direction, often indicated by their coordinates. This representation is key to visualizing and solving problems involving movement, forces, or any quantity with both direction and magnitude.

Vector Addition

Vector addition combines two or more vectors to form a resultant vector. This is typically done by adding corresponding components (e.g., x-components and y-components) of each vector. It is a fundamental operation used to calculate net displacements or to merge different vector quantities into a single vector result.

Transcript

00:01 This problem we're given a bunch of points and we are asked to find the sum of two vectors.

00:05 We're going to do this algebraically and we're going to do it geometrically.

00:09 So the first thing i'm going to do is graph the information given to me by each of these vectors.

00:15 So our first vector here is the vector from o to p.

00:19 Now o stands for just the origin here so you could think of that as a zero.

00:23 So that's going to be the point 0 ,0 and that's going to go to point p and point p was given to us here as negative 1 ,3.

00:30 So the vector from the origin to point p is that first vector o.

00:39 Okay same thing for qr.

00:41 So for qr we're going to go from the point q which is at 4 ,6.

00:46 2 ,3 ,4 ,5 ,6 and we're going to go to point r which is at 4 ,3.

00:53 So q to r order matters here right? so that's going from q down to r.

00:58 So this is going to be vector qr.

01:01 Okay now if i want to think about what movement or motion these vectors sort of indicate right then i can find that again graphically or algebraically.

01:14 So if i want to think about this graphically to begin with that first vector from o to p from the origin to point p moved to the left 1 and up 3.

01:26 So vector op would be denoted by negative 1 ,3.

01:32 Now your book might use brackets like that.

01:35 You could also write it using these pointy brackets negative 1 to 3.

01:39 Doesn't matter they're equivalent okay.

01:42 Q to r now is a little bit different.

01:44 Q to r doesn't have any left or right movement.

01:48 It just goes down.

01:50 So the x portion of this is going to be 0.

01:53 The y is going to be negative because we're heading down and that's going to be negative 3.

01:58 Okay now we could have found that algebraically if you want to think about the vector from the origin to point p.

02:07 What you would do is you would think about p as point turned sort of into a vector minus the 0 vector which is our origin.

02:17 Well p as a vector would be negative 1 ,3 and if we're subtracting from that the origin which is 0 ,0 i get negative 1 ,3.

02:26 Right exactly the same as what we got up top when we were looking at this algebraically.

02:30 Same thing for q to r.

02:33 So then you'd need to do r minus q.

02:35 It's always terminal point or ending point minus your initial point to where you started.

02:41 So here r as a vector would be 4 ,3 minus q which was that 4 ,6.

02:48 Subtracting corresponding components 4 minus 4 is 0...

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