Question

In each case, verify that the points \(P\) and \(Q\) lie on the line. $$ \begin{array}{ll} \text { a. } & x=3-4 t \quad P(-1,3,0), Q(11,0,3) \\ & y=2+t \\ & z=1-t \\ \text { b. } & x=4-t \quad P(2,3,-3), Q(-1,3,-9) \\ & y=3 \\ & z=1-2 t \end{array} $$

Short answer

Point Q lies on the line in both cases, while P does not in part (a), but does in part (b).

Step-by-step solution

Verify point P for part (a)

Substitute the coordinates of point \( P(-1, 3, 0) \) into the parametric equations: \[ x = 3 - 4t, \; y = 2 + t, \; z = 1 - t \] **For \(x = -1\):** \[-1 = 3 - 4t \rightarrow -4 = 4t \rightarrow t = -1\] **For \(y = 3\):** \[3 = 2 + t \rightarrow t = 1\] **For \(z = 0\):** \[0 = 1 - t \rightarrow t = 1\] With consistent parameters \( t = -1 \) and contradicting \( t = 1 \), point \( P \) does not lie on the line.

Verify point Q for part (a)

Substitute the coordinates of point \( Q(11, 0, 3) \) into the parametric equations: \[ x = 3 - 4t, \; y = 2 + t, \; z = 1 - t \] **For \(x = 11\):** \[11 = 3 - 4t \rightarrow 8 = -4t \rightarrow t = -2\] **For \(y = 0\):** \[0 = 2 + t \rightarrow t = -2\] **For \(z = 3\):** \[3 = 1 - t \rightarrow -t = 2 \rightarrow t = -2 \] All coordinates give consistent parameter \( t = -2 \), hence, point \( Q \) lies on the line.

Verify point P for part (b)

Substitute the coordinates of point \( P(2, 3, -3) \) into the parametric equations: \[ x = 4 - t, \; y = 3, \; z = 1 - 2t \] **For \(x = 2\):** \[2 = 4 - t \rightarrow t = 2\] **For \(y = 3\):** This equation is always satisfied since \( y = 3 \).**For \(z = -3\):** \[-3 = 1 - 2t \rightarrow -4 = -2t \rightarrow t = 2\] All coordinates give consistent parameter \( t = 2 \), hence, point \( P \) lies on the line.

Verify point Q for part (b)

Substitute the coordinates of point \( Q(-1, 3, -9) \) into the parametric equations: \[ x = 4 - t, \; y = 3, \; z = 1 - 2t \] **For \(x = -1\):** \[-1 = 4 - t \rightarrow -5 = -t \rightarrow t = 5\] **For \(y = 3\):** This equation is always satisfied since \( y = 3 \).**For \(z = -9\):** \[-9 = 1 - 2t \rightarrow -10 = -2t \rightarrow t = 5\] All coordinates give consistent parameter \( t = 5 \), hence, point \( Q \) lies on the line.

Key concepts

Line Verification

When tackling problems that involve verifying whether points lie on a line defined by parametric equations, line verification is an essential process. Parametric equations express the coordinates of points on a line in terms of a parameter, often denoted as \( t \). To confirm if a particular point is on the line, each coordinate of the point is substituted into its respective parametric equation.

This process involves using the point's coordinates to solve for \( t \). If solving the parametric equations with each coordinate results in the same parameter \( t \), the point indeed lies on the line.

• Begin by checking each coordinate one by one, substituting the point's coordinates into the parametric equations to evaluate \( t \).
• Compare the solutions for \( t \) from each coordinate. If they match across all coordinates, the point is on the line.
• If any coordinate yields a different \( t \), the point does not lie on the line.

In our exercise, points verified through this method clearly showcase the reliability of line verification in parametric equations.

Consistent Parameters

Consistent parameters refer to obtaining the same value of \( t \) when substituting different coordinates of a point into the parametric equations. This consistency signifies that the point indeed lies on the line.

To determine consistency:

• Substitute each coordinate of the point into its corresponding parametric equation and solve for \( t \).
• If the parameter \( t \) is consistent across all coordinates, the point lies on the line. For example, consistent parameters were found when verifying point Q in both parts (a) and (b) of the exercise.
• Conversely, if different coordinates provide different \( t \) values, inconsistency indicates the point is off the line, as seen with point P in part (a).

This concept ensures that the alignment of the parameter \( t \) across all equations is key to verifying the lie of points on a line.

Coordinate Substitution

Coordinate substitution is a technique used to check if a point lies on a line specified by parametric equations. It involves replacing the parameters in the equations with the known x, y, and z coordinates of a point.

Here's how to effectively perform coordinate substitution:

• Take the x, y, and z values of the given point and substitute them into the respective parametric equations.
• Solve each equation separately for \( t \).
• By comparing the resulting \( t \) values, assess whether they are consistent across all substituted equations.

For instance, in part (b) of the original exercise, substitution resulted in a consistent \( t = 2 \) for P and \( t = 5 \) for Q, confirming both points lie on the given line.
Simultaneously, the exercise highlights that only points with matching \( t \) validate their presence on the line based on the equations offered.