The task is to identify which parametric equations for lines pass through the point $(5, 7, 3)$ and are parallel to the vector $\langle 4, 2, 3 \rangle$.

Let's break this down:

1. Condition 1: The lines must contain the point $(5, 7, 3)$:

   • For each line, check if setting $t = 0$ gives the coordinates $(5, 7, 3)$.

2. Condition 2: The lines must be parallel to the vector $\langle 4, 2, 3 \rangle$:

   • In the parametric form of a line $x = x_0 + a t$, $y = y_0 + b t$, $z = z_0 + c t$, the vector $\langle a, b, c \rangle$ must match the direction vector $\langle 4, 2, 3 \rangle$.

Now, let's analyze each option:

First Equation:

$x = 6 + 2t, \quad y = 4 + 4t, \quad z = 8 + 6t$

• Direction vector: $\langle 2, 4, 6 \rangle$. This is not parallel to $\langle 4, 2, 3 \rangle$, since it is not a scalar multiple of $\langle 4, 2, 3 \rangle$.
• Therefore, this line is not parallel to the given vector.

Second Equation:

$x = 5 + t, \quad y = 7 + 2t, \quad z = 3 + 3t$

• Direction vector: $\langle 1, 2, 3 \rangle$. This is also not parallel to $\langle 4, 2, 3 \rangle$, as the $x$-component differs.
• Hence, this line is not parallel to the given vector.

Third Equation:

$x = 3 + 4t, \quad y = 3 + 2t, \quad z = -3 + 3t$

• Direction vector: $\langle 4, 2, 3 \rangle$, which is exactly the direction vector we're looking for.
• Check if the line contains the point $(5, 7, 3)$. Set $t$ such that:
  • $3 + 4t = 5$ → $t = \frac{1}{2}$
  • $3 + 2t = 7$ → $t = 2$ (Contradiction)
• Therefore, this line does not pass through the point $(5, 7, 3)$.

Fourth Equation:

$x = 4 + t, \quad y = 5 + 2t, \quad z = -3t$

• Direction vector: $\langle 1, 2, -3 \rangle$. This does not match $\langle 4, 2, 3 \rangle$.
• Hence, this line is not parallel to the given vector.

Conclusion:

None of the provided lines satisfy both conditions: passing through the point $(5, 7, 3)$ and being parallel to the vector $\langle 4, 2, 3 \rangle$.

Would you like a more detailed explanation on any part, or have further questions?

Related Questions:

1. How do you determine if two vectors are parallel?
2. What is the parametric form of a line in 3D space?
3. How can you find if a point lies on a given line?
4. What does it mean for two lines to be parallel in 3D space?
5. How do you find the equation of a line given a point and a direction vector?

Tip: When checking if a line passes through a point in parametric form, always substitute $t = 0$ to check if the starting point matches.