What Conditions on c, d, e Fill a Dashed Triangle Using Vectors u, v, w?

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Homework Statement

Under what restrictions on c,d,e will the combinations c*u+d*v+e*w fill in the dashed triangle?

Homework Equations

The Attempt at a Solution

I think c,d,e>=0 and bcos the triangle it's in a plane maybe remove one of c,d or e doing c+d+e=number?

 

[mutton]

The midpoint between u and v is (u + v)/2. Convince yourself by drawing a parallelogram. This fact can be used to find conditions on c, d such that the linear combinations cu + dv fill in the line segment between u and v.

A better way to think about this: The vector from u to v is v - u. Starting at u, we can vary the length of v - u to fill in the line segment. Consider u + t(v - u) for t in [0, 1].

To fill in the entire triangle including its inside, use the same sort of reasoning: Starting at u, we can vary the lengths of v - u and w - u under certain restrictions.

 

[Architeuthis Dux]

I would approach this problem by first defining what is meant by "filling in the dashed triangle." This could mean different things depending on the context, so it is important to clarify. For example, does it mean that the linear combination must result in a point within the triangle, on the boundary of the triangle, or any point in the plane that contains the triangle?

Assuming that "filling in the dashed triangle" means that the linear combination must result in a point within the triangle, the restrictions on c, d, and e would depend on the specific points u, v, and w that make up the triangle. For example, if the triangle is defined by the points (1,0), (0,1), and (1,1), then any combination of c, d, and e that results in a point within the triangle would satisfy the condition. However, if the triangle is defined by the points (0,0), (1,0), and (0,1), then the restrictions on c, d, and e would be more specific.

As for the equation c + d + e = number, this would only be applicable if the triangle is defined by a linear combination of two vectors, such as u and v. In this case, the linear combination c*u + d*v would span the entire plane containing the triangle, and e would be used to restrict the point to within the triangle. However, if the triangle is defined by a linear combination of three vectors, such as u, v, and w, then the equation c + d + e = number would not apply.

In conclusion, the restrictions on c, d, and e for the linear combination to fill in the dashed triangle would depend on the specific points u, v, and w that define the triangle. It is important to clarify what is meant by "filling in the dashed triangle" and to consider the specific context of the problem before determining the restrictions on c, d, and e.