Question

The equation of a line parallel to the vector \( 3 \hat{i} + \hat{j} + 2 \hat{k} \) and passing through the point \( (4, -3, 7) \) is:

• A. \( x = 4t + 3, y = -3t + 1, z = 7t + 2 \)
• B. \( x = 3t + 4, y = t + 3, z = 2t + 7 \)
• C. \( x = 3t + 4, y = -3, z = 2t + 7 \)
• D. \( x = 3t + 4, y = -3t + 1, z = 2t + 7 \)

Hint:

To find the equation of a line, use the parametric form \( x = x_0 + at, y = y_0 + bt, z = z_0 + ct \), where \( (x_0, y_0, z_0) \) is the given point and \( \langle a, b, c \rangle \) is the direction vector.

Answer 1

The Correct Option is B

The vector equation defines a line in 3D space. This line is parallel to the vector \( \mathbf{v} = 3\hat{i} + \hat{j} + 2\hat{k} \) and passes through the point \((x_0, y_0, z_0) = (4, -3, 7)\). The general vector form is \(\mathbf{r} = \mathbf{r_0} + t\mathbf{v}\).

Here, \(\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}\) represents any point on the line. The position vector of the given point is \(\mathbf{r_0} = 4\hat{i} - 3\hat{j} + 7\hat{k}\), and the direction vector is \(\mathbf{v} = 3\hat{i} + \hat{j} + 2\hat{k}\). Substituting these into the vector equation yields:

\[x\hat{i} + y\hat{j} + z\hat{k} = (4\hat{i} - 3\hat{j} + 7\hat{k}) + t(3\hat{i} + \hat{j} + 2\hat{k})\]

Separating this into components produces the parametric equations:

• \(x = 4 + 3t\)
• \(y = -3 + t\)
• \(z = 7 + 2t\)

The parametric representation of the line is thus:

\(x = 3t + 4, y = t - 3, z = 2t + 7\)