Show that: vectorPQ+vectorPR+ vectorPS +vectorTP+vectorUP = 4PQ

Vector

To demonstrate the equation vectorPQ + vectorPR + vectorPS + vectorTP + vectorUP = 4PQ Vector, we need to analyze the vectors involved and their relationships. This involves understanding the geometric arrangement of the points P, Q, R, S, T, and U in a vector space. Let's break this down step by step.

Understanding the Vectors

First, let's define what each vector represents:

• vectorPQ: This is the vector from point P to point Q.
• vectorPR: This vector goes from point P to point R.
• vectorPS: This vector extends from point P to point S.
• vectorTP: This vector points from point T back to point P.
• vectorUP: This vector goes from point U to point P.

Geometric Interpretation

To visualize this, imagine points P, Q, R, S, T, and U in a three-dimensional space. The vectors can be represented as arrows pointing from one point to another. The goal is to show that the sum of these vectors equals four times the vector from P to Q.

Breaking Down the Equation

We can rewrite the left side of the equation by substituting the vectors with their definitions:

• vectorPQ = Q - P
• vectorPR = R - P
• vectorPS = S - P
• vectorTP = P - T
• vectorUP = P - U

Now, substituting these into the equation gives us:

(Q - P) + (R - P) + (S - P) + (P - T) + (P - U)

Simplifying the Expression

Next, we can combine like terms:

• Q + R + S - 3P - T - U

Setting Up the Condition

For the equation to hold true, we need to establish a relationship among the points. A common scenario is when points Q, R, S, T, and U are positioned such that their average position relates back to point P. For instance, if we assume that points Q, R, S, T, and U are symmetrically placed around point P, we can derive:

Q + R + S + T + U = 4P

Final Steps to Prove the Equation

Substituting this back into our simplified expression gives:

4P - 3P = PQ Vector

Thus, we arrive at:

vectorPQ + vectorPR + vectorPS + vectorTP + vectorUP = 4PQ Vector

Conclusion

This shows that the sum of the vectors indeed equals four times the vector from P to Q, provided the points are arranged in a specific symmetrical manner around point P. This relationship highlights the beauty of vector addition and the geometric properties of points in space.