How to Deduce the Normal Vector from a Given Vector Equation?

In summary, an equation normal to a plane is a mathematical representation of a line perpendicular to the given plane. To find this equation, three non-collinear points on the plane are needed, and the cross product of two vectors in the plane is used to determine the direction. This equation has a unique solution and is used in various fields such as engineering, physics, and computer graphics. It helps calculate angles of incidence and reflection, determine slope, and create 3D models. It is important to note that the equation normal to a plane is different from the equation of the plane, as it determines the direction of the perpendicular line, not the points on the plane.
  • #1
acpower89
7
0
I'm having trouble finding the equation normal to the plane for part (ii).

The vector equation I got for part (i) is r=(1/2,1,2/3) + t(3/2,2,-2/3). How do you deduce the normal vector from this? The answer is the vector (9,12,-4). I've tried anything but don't know where they got this result.

Thanks
 

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  • #2
They used your (3/2,2,-2/3) and multiplied by 6 to clear the denominator.
 

FAQ: How to Deduce the Normal Vector from a Given Vector Equation?

What is an equation normal to a plane?

An equation normal to a plane is a mathematical representation of a line that is perpendicular to the given plane. It helps determine the direction and orientation of the plane in three-dimensional space.

How do you find the equation normal to a plane?

To find the equation normal to a plane, you need to know the coordinates of three non-collinear points on the plane. Then, you can use the cross product of two vectors in the plane to determine the direction of the normal line and use one of the points to find the equation of the line.

Can an equation normal to a plane have multiple solutions?

No, an equation normal to a plane has a unique solution. This is because a plane can only have one perpendicular line passing through it at a given point.

How is the equation normal to a plane used in real-world applications?

The equation normal to a plane is used in various fields such as engineering, physics, and computer graphics. It is used to calculate the angle of incidence and reflection of light rays on a surface, determine the slope of a ramp or inclined plane, and create 3D models and animations.

Is the equation normal to a plane the same as the equation of the plane?

No, the equation normal to a plane is a different mathematical representation than the equation of the plane. While the equation of the plane determines the points that lie on the plane, the equation normal to a plane determines the direction of the perpendicular line to the plane.

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