Express vector AO as a weighted combination of vectors AC and AE using the intersection point O of diagonals | Step-by-Step Solution
Problem
ABCD is a parallelogram. AE = 1/3 AD, BC = 4BF. O is the intersection of DF and EC. How do you use AE and AC to express AO through so: AO = (EO / (EO + CO)) * AC + (CO / (EO + CO)) * AE
🎯 What You'll Learn
- Understand vector decomposition techniques
- Learn to express vectors as weighted combinations
- Analyze geometric relationships between vectors
Prerequisites: Basic vector operations, Parallelogram properties, Vector coordinate representation
💡 Quick Summary
Hi there! This is a really interesting vector algebra problem that involves finding intersection points in parallelograms - it's like detective work with vectors! The key insight here is that when two lines intersect, you can describe each line using parametric equations and then find where those equations are equal. Have you considered setting up parametric equations for both lines DF and EC using parameters like t and s? You'll want to think about how to express vectors in terms of your base vectors (like AB and AD), and remember that in a parallelogram, opposite sides are equal vectors. Once you set up those parametric equations, the intersection point O will be where both equations describe the same point, so you can set them equal and solve for your parameters. Give it a try by first expressing all the relevant points (like F and E) in terms of your base vectors using the given conditions!
Step-by-Step Explanation
Hey there! This is a vector algebra problem that combines properties of parallelograms with intersection points.
What We're Solving:
We need to express vector AO as a weighted combination of vectors AC and AE, where O is the intersection of lines DF and EC in parallelogram ABCD, with the given conditions AE = (1/3)AD and BC = 4BF.The Approach:
The key insight here is that when two lines intersect, we can express the intersection point using parametric equations for each line. We'll:- 1. Set up parametric equations for lines DF and EC
- 2. Find where they intersect (point O)
- 3. Express AO in terms of the given vectors
Step-by-Step Solution:
Step 1: Set up our coordinate system using vectors Let's use AB and AD as our base vectors. In a parallelogram:
- AC = AB + AD (diagonal vector)
- AE = (1/3)AD (given condition)
- BC = AD (opposite sides of parallelogram are equal)
- So BF = (1/4)AD
- Therefore: AF = AB + BF = AB + (1/4)AD
Step 4: Set up parametric equation for line EC Any point on line EC can be written as: AP = AE + s(AC - AE) AP = (1/3)AD + s(AB + AD - (1/3)AD) AP = (1/3)AD + s(AB + (2/3)AD)
Step 5: Find intersection point O At intersection, both parametric equations are equal: AD + t(AB - (3/4)AD) = (1/3)AD + s(AB + (2/3)AD)
Expanding and collecting terms: AB: t = s AD: 1 - (3t/4) = 1/3 + (2s/3)
Step 6: Solve for parameters Since t = s, substitute into the AD equation: 1 - (3t/4) = 1/3 + (2t/3) 1 - 1/3 = (3t/4) + (2t/3) 2/3 = (9t + 8t)/12 = 17t/12
Therefore: t = 8/17
Step 7: Express AO AO = (1/3)AD + (8/17)(AB + (2/3)AD) AO = (1/3)AD + (8/17)AB + (16/51)AD AO = (8/17)AB + (17/51 + 16/51)AD AO = (8/17)AB + (33/51)AD AO = (8/17)AB + (11/17)AD
Step 8: Express in terms of AC and AE Now we can write this as: AO = (8/17)(AB + AD) + (3/17)AD AO = (8/17)AC + (3/17)AD AO = (8/17)AC + (9/17)AE
The Answer:
AO = (8/17)AC + (9/17)AEMemory Tip:
When finding intersection points with vectors, always remember: Set up parametric equations for both lines, then solve where they're equal! The magic happens when you equate the coefficients of your base vectors. 🌟Great job working through this complex problem - vector intersections can be tricky, but you're building really strong problem-solving skills!
⚠️ Common Mistakes to Avoid
- Misinterpreting the weighted coefficients
- Incorrect application of vector addition rules
- Failing to understand the geometric constraints
This explanation was generated by AI. While we work hard to be accurate, mistakes can happen! Always double-check important answers with your teacher or textbook.
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📷 Problem detected:
Solve: 2x + 5 = 13
Step 1:
Subtract 5 from both sides...
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