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 aa, bb, and cc such that

r4=ar1+br2+cr3,\vec{r}_4 = a \vec{r}_1 + b \vec{r}_2 + c \vec{r}_3,

we need to express r4\vec{r}_4 as a linear combination of r1\vec{r}_1, r2\vec{r}_2, and r3\vec{r}_3.

Given: r1=2i^j^+k^,\vec{r}_1 = 2\hat{i} - \hat{j} + \hat{k}, r2=i^+3j^2k^,\vec{r}_2 = \hat{i} + 3\hat{j} - 2\hat{k}, r3=2i^+j^3k^,\vec{r}_3 = -2\hat{i} + \hat{j} - 3\hat{k}, r4=3i^+2j^+5k^.\vec{r}_4 = 3\hat{i} + 2\hat{j} + 5\hat{k}.

Step 1: Set up the equation in component form

Since r4=ar1+br2+cr3\vec{r}_4 = a \vec{r}_1 + b \vec{r}_2 + c \vec{r}_3, we can write this equation in terms of the i^\hat{i}, j^\hat{j}, and k^\hat{k} components.

  1. For i^\hat{i}-components: 3=2a+b2c.3 = 2a + b - 2c.

  2. For j^\hat{j}-components: 2=a+3b+c.2 = -a + 3b + c.

  3. For k^\hat{k}-components: 5=a2b3c.5 = a - 2b - 3c.

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{

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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