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 .

To find the scalars \( a \), \( b \), and \( c \) such that 

\[
\vec{r}_4 = a \vec{r}_1 + b \vec{r}_2 + c \vec{r}_3,
\]

we need to express \( \vec{r}_4 \) as a linear combination of \( \vec{r}_1 \), \( \vec{r}_2 \), and \( \vec{r}_3 \).

Given:
\[
\vec{r}_1 = 2\hat{i} - \hat{j} + \hat{k},
\]
\[
\vec{r}_2 = \hat{i} + 3\hat{j} - 2\hat{k},
\]
\[
\vec{r}_3 = -2\hat{i} + \hat{j} - 3\hat{k},
\]
\[
\vec{r}_4 = 3\hat{i} + 2\hat{j} + 5\hat{k}.
\]

### Step 1: Set up the equation in component form

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

1. **For \( \hat{i} \)-components:**
   \[
   3 = 2a + b - 2c.
   \]

2. **For \( \hat{j} \)-components:**
   \[
   2 = -a + 3b + c.
   \]

3. **For \( \hat{k} \)-components:**
   \[
   5 = a - 2b - 3c.
   \]

This gives us a system of linear equations:

\[
\begin{cases}
2a + b - 2c = 3, \\
-a + 3b + c = 2, \\
a - 2b - 3c = 5.
\end{cases}
\]

### 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{

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.

Learn how to find scalars a, b, and c such that the vector r⃗1 is expressed as a linear combination of r⃗1, r⃗2, and r⃗3. This problem involves solving a system of linear equations to express one vector as a sum of others. Suitable for students in Grades 11-12 studying vector algebra.

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