9.1 The Intersection of a Line with a Plane and the Intersention of Two Lines

Question 1

how to show that the equation of a line in parametric form lies on a plane in cartesian form?

Answer 1

1. substitute the parametric equation into the plane equation
2. if both sides of the equation are equal, this proves the prametric equation is also on the plane

Question 2

how to show that the equation of a line in vector equation form lies on a plane in cartesian form?

Answer 2

1. find the parametric equation of the line
2. substuitute the parametric equation from the line into the plane equation
3. if both sides equal zero, this proves all point on the line lie on the plane

Question 3

how to verify that the equation in vector equation form of a line lies on a plane?

Answer 3

1. show that the direction vector of the line and the normal of the plane meet and right angles by taking the dot product
2. substitute the point into the plane equation, and if both sides do not equal this proves the point does not lie on the line

Question 4

how to determine the point of intersection between a line in vector equation form and the plane in cartesian form?

Answer 4

1. find the parametric equations of the line
2. substitute the parametric equation into the plane to find the value of the direction vector variable
3. plug the variable back into the vector equation to find the point of intersection

Question 5

how to determine the point of intersection between pairs lines?

Answer 5

1. find the parametric equations of both lines
2. set the parametric equation equal to each other
3. Isolate for one variable
4. substitute equation for the variable into another equation to isolate for a variable
5. substitute the variable into one of the equation to find the other variable
6. substitute variable back into original lines equation and see if the points in both are the same. if they are the same. then that would be the point of intersection between the two pairs of lines

Question 6

how to determine if a pair of lines both in vector equation form are skew?

Answer 6

1. write the parametric equations of both the lines
2. set the parametric equations equal to each other
3. Isolate for one variable
4. substitute equation for the variable into another equation to isolate for another variable
5. substitute the variable into one of the equation to find the other variable
6. substitute variable back into original lines equation and see if the points in both are the same. if they are not both the same, then the lines are skew

Question 7

if a line in vector equation form intersects with the z-axis at the point (0,0,q) how do you find the value of q?

Answer 7

1. write the parametric equation of the line
2. set one the parametric equations equal to the 0 coordinate in the point
3. Isolate for the variable
4. substitute variable into the equation to find the point

NOTE: the x and y coordinates will be zero because the question highlights that the line intersect with the z-axis

Question 8

how to show that a point lies on a line in vector equation form?

Answer 8

1. write the parametric equations of the line
2. set the parametric equation equal to the appropriate coordinate in the point
3. isolate for the variable in the line equation
4. substitute the value of the variable in the original parametric equations and see if they produce the same coordinates as the point given

Question 9

what does proving that the point lies on a line and two lines having the same direction vector prove?

Answer 9

the proves that both the lines are coincident

Question 10

when given a line in vector equation form and another line in vector equation form with a variable in the direction vector, how do you determine the value of k?

Answer 10

1. write the parametric equations of the line
2. set the parametric equations equal to each other
3. Isolate for one variable
4. substitute equation for the variable into one of the set equal to parametric equations to isolate for another variable
5. substitute the variable into one of the equation to find the other variable
6. plug the variables into one of the set equal to parametric equations to solve and isolate for the value of k

Question 11

given that a line in vector equation form intersects the xz and yz coordinates plane at points A and B, how do you determine the length of the line segment?

Answer 11

1. Write the coordinates of point A and B
   A(x,0,z) and B(0,y,z)
2. write the parametric equations of the line
3. set the y parametric equation equal to the y component of the point A to find the value of the variable
4. plug the variable back into the vector line equation to find the point
5. set the x parametric equation equal to the x component of the B point to find the value of the variable
6. Label the points x1,y1,z1 and x2,y2,z2
7. and the distance between the two points using the distance between two points formula

d=square root (x2,x1)^2+(y2-y1)^2+(z2-z1)^2