Find the vector equation of a line through the point (-3, 2, -4) and is parallel to the vector ` 3hati + 2hatj+ 4hatk`.

Answer

To find the vector equation of a line that passes through the point \((-3, 2, -4)\) and is parallel to the vector \(3\hat{i} + 2\hat{j} + 4\hat{k}\), we can follow these steps:

### Step 1: Identify the point and direction vector
We are given the point \(A(-3, 2, -4)\) and the direction vector \( \mathbf{b} = 3\hat{i} + 2\hat{j} + 4\hat{k} \).

### Step 2: Write the vector form of the point
The position vector of point \(A\) can be written as:
\[ \mathbf{a} = -3\hat{i} + 2\hat{j} - 4\hat{k} \]

### Step 3: Write the general equation of the line
The vector equation of a line can be expressed in the form:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
where \(\mathbf{r}\) is the position vector of any point on the line, \(\mathbf{a}\) is the position vector of a known point on the line, \(\mathbf{b}\) is the direction vector, and \(\lambda\) is a scalar parameter.

### Step 4: Substitute the values into the equation
Substituting the values of \(\mathbf{a}\) and \(\mathbf{b}\) into the equation:
\[ \mathbf{r} = (-3\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} + 4\hat{k}) \]

### Step 5: Simplify the equation
Distributing \(\lambda\) in the equation gives:
\[ \mathbf{r} = -3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(3\hat{i} + 2\hat{j} + 4\hat{k}) \]
This can be rewritten as:
\[ \mathbf{r} = (-3 + 3\lambda)\hat{i} + (2 + 2\lambda)\hat{j} + (-4 + 4\lambda)\hat{k} \]

### Final Vector Equation
Thus, the vector equation of the line is:
\[ \mathbf{r} = (-3 + 3\lambda)\hat{i} + (2 + 2\lambda)\hat{j} + (-4 + 4\lambda)\hat{k} \]

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