Consider the plane 5 x-y+7 z=21 (a) Find a point on the x -axis on this plane. (b) Find two othe

step 1 Hello there. So for this exercise we have the following plane. I'm going to call it pi, but this

Consider the plane 5 x-y+7 z=21 (a) Find a point on the x...

Key Concepts

Normal Vector

The normal vector to a plane is a vector that is perpendicular to the plane and can be directly obtained from the coefficients of x, y, and z in the plane's equation. This vector is crucial in determining the orientation of the plane and is often used in calculations involving distances, projections, and angles between planes.

Parallel Vector to a Plane

A vector parallel to a plane lies entirely within the plane and is orthogonal to the plane's normal vector. One common method to find such a vector is to choose specific values for two coordinates and solve for the third, ensuring that the resulting vector (formed by the difference between points) has a dot product of zero with the normal vector.

Equation of a Plane

A plane in three-dimensional space is typically described by a linear equation in x, y, and z. This equation represents all the points (x, y, z) that lie on the plane, and manipulating this equation can help identify various properties of the plane such as intercepts or intersections with other geometrical objects.

Intercepts on Coordinate Axes

Finding intercepts involves setting the coordinates corresponding to the other axes to zero. For example, to find the x-axis intercept of a plane, the y and z coordinates are set to zero. This method is a specific instance of finding points that lie on the plane by reducing the number of variables in the plane's equation.

Transcript

0:00 Hello there.

00:01 So for this exercise we have the following plane.

00:06 I'm going to call it pi, but this defined by the following equation, 5x minus y plus 7z equals to 21.

00:14 So this defined, this equation of a plane.

00:17 And the first thing that we need to do is find a point of this plane that lies on the x -axis.

00:26 So what is the meaning that lies on the x -axis? well, so technically that translates into equations that x is equal to zero.

00:36 If x is equal to zero, then the equation of our plane becomes minus y plus 7 z equals to 21.

00:49 And from here, we have, well, we have two variables and one equation, so we can give some arbitrary value either for z or y.

00:59 So let's suppose that z is equal to 23.

01:10 Then this equation implies that y it's going to be equals to 0.

01:17 So here we have the three components of our point.

01:22 So that point that lies on the x -axis, i'm going to call it px, is a point zero zero three.

01:35 Now we need to find other two points.

01:40 So let's say b1 and b2 that also lies on the plane.

01:46 So how to find these these points? well, we repeat like the same procedure as before.

01:55 We have one equation, 5x minus y plus 7 z equals to 21.

02:08 Okay, so we have 5x minus y plus 7 z equals to 21 and we need 5x minus y plus 7 z equals to 21 and we need to give two values for two of the variables that we have here.

02:23 It could be whenever you want.

02:27 It could be x and y, x and z, y and z, or yeah, any combination, it doesn't matter...

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