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 equation of a line in parametric form is: \[ x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct \] where \( (x_0, y_0, z_0) \) is the point through which the line passes, and \( \langle a, b, c \rangle \) is the direction vector of the line. Here, the direction vector is \( \langle 3, 1, 2 \rangle \) and the point is \( (4, -3, 7) \). Therefore, the parametric equations are: \[ x = 3t + 4, \quad y = t - 3, \quad z = 2t + 7 \]