Question
Vectors u, v, and w are drawn that create a right triangle.3 vectors drawn that create a right triangle. Vector v is vertical pointing down. Vector u is horizontal pointing right. The initial points of v and u coincide to create the right angle. Vector w is diagonal and joins the two terminal sides of vectors u and v. Vector w is pointing upwards at the terminal side of vector u.
Question
Use the image to answer the question.
Which mathematical statement matches the vector operation shown in the geometric representation?
(1 point)
Responses
bold v plus bold w equals bold u
Image with alt text: bold v plus bold w equals bold u
bold u plus bold v equals bold w
Image with alt text: bold u plus bold v equals bold w
bold w plus bold u equals bold v
Image with alt text: bold w plus bold u equals bold v
bold w minus bold v equals bold u
Image with alt text: bold w minus bold v equals bold u
Question
Use the image to answer the question.
Which mathematical statement matches the vector operation shown in the geometric representation?
(1 point)
Responses
bold v plus bold w equals bold u
Image with alt text: bold v plus bold w equals bold u
bold u plus bold v equals bold w
Image with alt text: bold u plus bold v equals bold w
bold w plus bold u equals bold v
Image with alt text: bold w plus bold u equals bold v
bold w minus bold v equals bold u
Image with alt text: bold w minus bold v equals bold u
Answers
GPT-4o mini
In the given geometric representation of vectors forming a right triangle, we can analyze the relationship between the vectors u, v, and w.
Since vector v is vertical and pointing down, vector u is horizontal and pointing right, and vector w is the diagonal connecting the ends of u and v (and pointing upwards), we can interpret the vectors in the following way:
1. Vector u (horizontal, right) represents one leg of the triangle.
2. Vector v (vertical, down) represents the other leg of the triangle.
3. Vector w (the hypotenuse) connects the terminal points of u and v.
According to the triangle's properties, we know that:
- The sum of vectors u and v should give vector w, which means:
\[
\mathbf{u} + \mathbf{v} = \mathbf{w}
\]
However, since vector v is directed downwards, it is being subtracted to get to the top of vector w that points up towards the terminal side of u.
Thus, from the geometric interpretation, the relationship described by the vectors can also be written as:
\[
\mathbf{w} - \mathbf{v} = \mathbf{u}
\]
From the given options, this corresponds to:
**bold w minus bold v equals bold u**
So, the correct answer is: **bold w minus bold v equals bold u**.
Since vector v is vertical and pointing down, vector u is horizontal and pointing right, and vector w is the diagonal connecting the ends of u and v (and pointing upwards), we can interpret the vectors in the following way:
1. Vector u (horizontal, right) represents one leg of the triangle.
2. Vector v (vertical, down) represents the other leg of the triangle.
3. Vector w (the hypotenuse) connects the terminal points of u and v.
According to the triangle's properties, we know that:
- The sum of vectors u and v should give vector w, which means:
\[
\mathbf{u} + \mathbf{v} = \mathbf{w}
\]
However, since vector v is directed downwards, it is being subtracted to get to the top of vector w that points up towards the terminal side of u.
Thus, from the geometric interpretation, the relationship described by the vectors can also be written as:
\[
\mathbf{w} - \mathbf{v} = \mathbf{u}
\]
From the given options, this corresponds to:
**bold w minus bold v equals bold u**
So, the correct answer is: **bold w minus bold v equals bold u**.