Math Problem Statement
If r⃗1 = 2î− ĵ+ k̂, r⃗2 = î+ 3ĵ− 2k̂, r⃗3 = −2î+ ĵ− 3k̂ and r⃗4 = 3î+ 2ĵ+ 5k̂, find
scalars a, b, c such that r⃗1 = ar⃗1 + b r⃗2 + c r⃗3 .
Solution
To find the scalars , , and such that
we need to express as a linear combination of , , and .
Given:
Step 1: Set up the equation in component form
Since , we can write this equation in terms of the , , and components.
-
For -components:
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For -components:
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For -components:
This gives us a system of linear equations:
2a + b - 2c = 3, \\ -a + 3b + c = 2, \\ a - 2b - 3c = 5. \end{cases}$$ ### Step 2: Solve the system of equations Let's solve this system step by step. 1. From the first equation: $$2a + b - 2c = 3.$$ 2. From the second equation: $$-a + 3b + c = 2.$$ 3. From the third equation: $$a - 2b - 3c = 5.$$ Let's solve for $$a$$, $$b$$, and $$c$$ using substitution or elimination. #### Solving by elimination: 1. Add the first and third equations to eliminate $$c$$: $$(2a + b - 2c) + (a - 2b - 3c) = 3 + 5.$$ Simplifying: $$3a - b - 5c = 8. \quad \text{(Equation 4)}$$ 2. Now add the second and third equations to eliminate $$c$$: $$(-a + 3b + c) + (a - 2b - 3c) = 2 + 5.$$ Simplifying: $$b - 2c = 7. \quad \text{(Equation 5)}$$ Now we have two new equations: $$3a - b - 5c = 8,$$ $$b - 2c = 7.$$ #### Substitute $$b = 7 + 2c$$ into Equation 4: From Equation 5, we get: $$b = 7 + 2c.$$ Substitute this into Equation 4: $$3a - (7 + 2c) - 5c = 8.$$ Simplify: $$3a - 7 - 7c = 8,$$ $$3a - 7c = 15,$$ \[ a - \frac{Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Linear Combinations
Systems of Linear Equations
Formulas
Vector equation: r⃗ = a r⃗1 + b r⃗2 + c r⃗3
System of linear equations
Theorems
Linear dependence and independence
Solving systems of linear equations
Suitable Grade Level
Grades 11-12
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