Understanding the Intersection of Planes

The video tutorial explains how to find the equation of the line formed by the intersection of two planes. It covers the identification of normal vectors for each plane, the use of cross product to determine the directional vector, and the algebraic method to find a point on the line. Finally, it formulates the line equation in parametric form and as a vector-valued function.

What is the normal vector to the plane given by the equation x + 2y - 4z = 16?

• (1, 2, -4)

• (2, -1, 3)

• (1, -2, 4)

• (2, 1, -3)

Which of the following is necessary to write the parametric equations of a line in space?

• A normal vector and a direction vector

• A point on the line and a normal vector

• A point on the line and a direction vector

• Two points on the line

What is the result of the cross product of two normal vectors?

• A point on the line

• The normal vector to the line

• The directional vector of the line

• The equation of the line

How do you find a point on the line of intersection of two planes?

• By finding the midpoint of the planes

• By solving the system of equations derived from the planes

• By taking the average of the normal vectors

• By using the cross product of the normal vectors

What is the value of z when x is set to zero in the system of equations derived from the planes?

• 16

• 14

• 12

• 18

What is the y-coordinate of the point on the line of intersection when x is zero?

• 36

• 28

• 14

• 42

What are the parametric equations for the line of intersection?

• x(t) = 2t, y(t) = 36 - 5t, z(t) = 14 - 11t

• x(t) = 2t, y(t) = 36 - 11t, z(t) = 14 - 5t

• x(t) = 2t, y(t) = 36 + 11t, z(t) = 14 + 5t

• x(t) = 2t, y(t) = 11t - 36, z(t) = 5t - 14

Which of the following is an alternative way to express the line of intersection?

• As a matrix

• As a scalar

• As a vector valued function

• As a single equation

What is the x-component of the vector valued function for the line of intersection?

• 2t

• 36 - 11t

• 14 - 5t

• 11t - 36

What is the most common way to write the equation of a line of intersection?

• As a single equation

• As a vector valued function

• As a set of parametric equations

• As a matrix