Verify if the point having positions vector ` 4 hati-11 hatj +2 hatk ` lies on the line ` overset (-)r =( 6 hati -4 hatj +5hatk ) +mu (2hati +7hatj +3hatk ) `

Answer

To verify if the point with position vector \( \mathbf{r_0} = 4 \hat{i} - 11 \hat{j} + 2 \hat{k} \) lies on the line given by the vector equation

\[ \mathbf{r} = (6 \hat{i} - 4 \hat{j} + 5 \hat{k}) + \mu (2 \hat{i} + 7 \hat{j} + 3 \hat{k}), \]

we need to find a value of \( \mu \) such that

\[ \mathbf{r} = \mathbf{r_0}. \]

### Step 1: Set up the equation
We can express the equation of the line in terms of its components:

\[ \mathbf{r} = (6 + 2\mu) \hat{i} + (-4 + 7\mu) \hat{j} + (5 + 3\mu) \hat{k}. \]

### Step 2: Equate components
Now, we will equate the components of \( \mathbf{r} \) with \( \mathbf{r_0} \):

1. For the \( \hat{i} \) component:
\[ 6 + 2\mu = 4. \]

2. For the \( \hat{j} \) component:
\[ -4 + 7\mu = -11. \]

3. For the \( \hat{k} \) component:
\[ 5 + 3\mu = 2. \]

### Step 3: Solve for \( \mu \)
Now we will solve each equation for \( \mu \).

**From the \( \hat{i} \) component:**
\[ 6 + 2\mu = 4 \implies 2\mu = 4 - 6 \implies 2\mu = -2 \implies \mu = -1. \]

**From the \( \hat{j} \) component:**
\[ -4 + 7\mu = -11 \implies 7\mu = -11 + 4 \implies 7\mu = -7 \implies \mu = -1. \]

**From the \( \hat{k} \) component:**
\[ 5 + 3\mu = 2 \implies 3\mu = 2 - 5 \implies 3\mu = -3 \implies \mu = -1. \]

### Step 4: Conclusion
Since all three components give the same value \( \mu = -1 \), we conclude that the point \( \mathbf{r_0} = 4 \hat{i} - 11 \hat{j} + 2 \hat{k} \) lies on the line defined by the given vector equation.