Question

in the above diagram, a student goes from home to the park, then to the movie theater, and finally to the stadium. each stage of the journey is shown as a separate vector. when all of the vectors are added together, which direction is the sum of all the vectors? options: east, south, west, north.

Explanation:

Step1: Analyze each vector

- Home to Park: Let's assume the direction from Home to Park is some vector (say, vector \(\vec{v_1}\)).
- Park to Movie Theater: Vector \(\vec{v_2}\) (from Park to Movie Theater).
- Movie Theater to Stadium: Vector \(\vec{v_3}\) (from Movie Theater to Stadium).

But we can also think in terms of the resultant vector from Home to Stadium directly (since vector addition is commutative and associative, the sum of Home→Park, Park→Movie Theater, Movie Theater→Stadium is equal to Home→Stadium).

Looking at the diagram, the path from Home to Park, then Park to Movie Theater, then Movie Theater to Stadium. But if we consider the net displacement (sum of vectors), we can see the direction from Home to Stadium. From the diagram, the Movie Theater to Stadium is a vector, and when we add all the vectors (Home→Park, Park→Movie Theater, Movie Theater→Stadium), the resultant should be the same as Home→Stadium? Wait, no, wait. Wait, the student goes Home→Park (vector 1), Park→Movie Theater (vector 2), Movie Theater→Stadium (vector 3). So the sum of vectors is \(\vec{v_1}+\vec{v_2}+\vec{v_3}\). But let's look at the directions. Let's assume the compass: N (up), S (down), E (right), W (left).

Home to Park: Let's say Home is at some point, Park is to the northeast? Wait, no, the diagram: Home has an arrow to Park (so Home→Park is a vector). Then Park has an arrow to Movie Theater (Park→Movie Theater). Then Movie Theater has an arrow to Stadium (Movie Theater→Stadium). Now, if we add these vectors, the resultant should be from Home to Stadium. Wait, but maybe the key is that the sum of Home→Park, Park→Movie Theater, Movie Theater→Stadium is equal to Home→Stadium? Wait, no, because vector addition is \(\vec{Home→Park} + \vec{Park→Movie Theater} + \vec{Movie Theater→Stadium} = \vec{Home→Stadium}\). Wait, but in the diagram, the arrow from Movie Theater to Stadium: let's see the direction. The compass is N (top), S (bottom), W (left), E (right). So Movie Theater to Stadium: the arrow is going down and to the right? Wait, no, the diagram: Movie Theater is top left, Stadium is bottom right? Wait, no, the labels: Movie Theater is top left, Park is top middle, Home is middle left, Stadium is bottom middle. Wait, the arrow from Home to Park: Home (middle left) to Park (top middle) – so that's a vector going up and right? Then Park to Movie Theater: Park (top middle) to Movie Theater (top left) – that's a vector going left. Then Movie Theater to Stadium: Movie Theater (top left) to Stadium (bottom middle) – that's a vector going down and right. Now, let's add these vectors:

1. Home→Park: let's represent as (x1, y1) where x is east (right), y is north (up). So from Home to Park: moving right (east) and up (north).
2. Park→Movie Theater: moving left (west) – so x decreases, y same?
3. Movie Theater→Stadium: moving down (south) and right (east).

Now, let's see the net x (east-west) and net y (north-south):

- East-West: (Home→Park east) + (Park→Movie Theater west) + (Movie Theater→Stadium east). If Home→Park east and Movie Theater→Stadium east, and Park→Movie Theater west. But maybe the key is that the sum of Park→Movie Theater (west) and Movie Theater→Stadium (east and south) and Home→Park (north and east) – but maybe a simpler way: the resultant vector is Home to Stadium. Wait, no, the question is the sum of all vectors (Home→Park, Park→Movie Theater, Movie Theater→Stadium) – so the resultant is from Home to Stadium? Wait, no, vector addition: \(\vec{AB} + \vec{BC} + \vec{CD} = \vec{AD}\). So Home (A) → Park (B) → Movie Theater (C) → Stadium (D). So the sum is \(\vec{AD}\) (Home to Stadium). Now, looking at the direction of Home to Stadium: from Home (middle left) to Stadium (bottom middle) – that's a vector going down (south) and maybe same east-west? Wait, no, the arrow from Movie Theater to Stadium: Movie Theater is top left, Stadium is bottom middle. So Movie Theater to Stadium: down (south) and right (east). Park to Movie Theater: left (west). Home to Park: up (north) and right (east). Let's cancel out:

- East: Home→Park east + Movie Theater→Stadium east - Park→Movie Theater west (since west is negative east). If Home→Park east and Movie Theater→Stadium east are equal to Park→Movie Theater west, then east components cancel.
- North: Home→Park north.
- South: Movie Theater→Stadium south.

Wait, maybe the key is that the sum of the vectors: Home→Park (let's say north-east), Park→Movie Theater (west), Movie Theater→Stadium (south-east). But when we add them, the west and east (from Home→Park and Movie Theater→Stadium) might cancel, and the north (from Home→Park) and south (from Movie Theater→Stadium) – but maybe the resultant is south? Wait, no, let's look at the options: east, south, west, north.

Wait, maybe the diagram is such that Home to Park is a vector, Park to Movie Theater is a vector opposite to Home to Park? No, the arrow from Home to Park is towards Park, Park to Movie Theater is towards Movie Theater (left), Movie Theater to Stadium is towards Stadium (down-right). Wait, maybe the sum of the vectors is the same as Home to Stadium, and the direction of Home to Stadium is south? Wait, no, let's think again.

Alternatively, maybe the vectors are:

- Home to Park: let's say vector \(\vec{v}\) (direction, say, north-east)
- Park to Movie Theater: vector \(-\vec{v}\) (opposite direction, north-west? No, Park to Movie Theater is left, so west)
- Movie Theater to Stadium: vector \(\vec{w}\) (south-east)

But maybe the key is that the sum of Home→Park and Park→Movie Theater is Home→Movie Theater (since \(\vec{Home→Park} + \vec{Park→Movie Theater} = \vec{Home→Movie Theater}\)). Then add \(\vec{Movie Theater→Stadium}\) to get \(\vec{Home→Stadium}\). Now, looking at the direction of Home→Movie Theater: Home is middle left, Movie Theater is top left – so that's north (same x, y increases). Then Movie Theater→Stadium: Movie Theater (top left) to Stadium (bottom middle) – that's south and east. So Home→Movie Theater is north, then Movie Theater→Stadium is south and east. So the resultant is east and (north - south). If north and south are equal, then east? No, maybe not. Wait, the options are east, south, west, north.

Wait, maybe the diagram is simpler: the three vectors are Home→Park, Park→Movie Theater, Movie Theater→Stadium. When you add them, the resultant is Home→Stadium. Now, looking at the arrow from Movie Theater to Stadium: it's going down (south) and maybe the same as the other vectors. Wait, maybe the answer is south? No, wait, let's check the compass. The compass has N (top), S (bottom), W (left), E (right). So if the resultant vector (sum of all vectors) is going south, then the direction is south. Wait, maybe I made a mistake. Let's re-express:

Vector 1: Home to Park (let's say direction: north-east)
Vector 2: Park to Movie Theater (direction: west)
Vector 3: Movie Theater to Stadium (direction: south-east)

Now, add the x-components (east is positive, west is negative):

- Vector 1 x: +a (a > 0)
- Vector 2 x: -b (b > 0)
- Vector 3 x: +c (c > 0)

If a + c = b, then x-component is 0.

Y-components (north is positive, south is negative):

- Vector 1 y: +d (d > 0)
- Vector 2 y: 0 (since it's west, same y)
- Vector 3 y: -e (e > 0)

If d = e, then y-component is 0. But that can't be. Wait, maybe the diagram is such that the sum of the vectors is the same as Home to Stadium, and the direction of Home to Stadium is south. Wait, the arrow from Movie Theater to Stadium is going down (south), so maybe the resultant is south.

Wait, the options are east, south, west, north. Let's think of the vectors:

Home to Park: let's say up (north) and right (east)
Park to Movie Theater: left (west) – so cancels the east from Home to Park
Movie Theater to Stadium: down (south) – so cancels the north from Home to Park, and maybe some east? No, if Park to Movie Theater cancels the east from Home to Park, then the remaining is south from Movie Theater to Stadium. So the resultant is south.

Yes, that makes sense. So the sum of the vectors is in the south direction.

Answer:

south (the option with "south")